Monetary Policy & Credit Easing pt. 7: R Econometrics Tests

In post 6 we introduced some econometrics code that will help those working with time-series to gain asymptoticly efficient results.  In this post we look at the different commands and libraries necessary for testing our assumptions and such.

Testing our Assumptions and Meeting the Gauss-Markov Theorem

In this section we will seek to test and verify the assumptions of the simple linear regression model.  These assumptions are laid out as follows and are extracted from Hill, Griffiths and Lim 2008:

SR1. The value of y, for each value of x, is
y= ß_{1}+ß_{2}x+µ
SR2. The expected value of the random error µ is
E(µ)=0
which is equivalent to assuming
E(y)= ß_{1}+ß_{2}x
SR3. The variance of the random error µ is
var(µ)=sigma^2 = var(y)
The random variables y and µ have the same variance because they differ only by a constant.
SR4. The covariance between any pair of random errors µ_{i} and µ_{j} is
cov(µ_{i}, µ_{j})=cov(y_{i},y_{j})=0
SR5. The variable x is not random and must take at least two different values.
SR6. The values of µ are normally distributed about their mean
µ ~ N(0, sigma^2)
if the y values are normally distributed and vice-versa 

Central to this topics objective is meeting the conditions set forth by the Gauss-Markov Theorem.  The Gauss-Markov Theorem states that if the error term is stationary and has no serial correlation, then the OLS parameter estimate is the Best Linear Unbiased Estimate or BLUE, which implies that all other linear unbiased estimates will have a larger variance. An estimator that has the smallest possible variance is called an "efficient" estimator.  In essence, the Gauss-Markov theorem states that the error term must have no structure; the residual levels must exhibit no trend and the variance must be constant through time.

When the error term in the regression does not satisfy the assumptions set forth by Gauss-Markov, OLS is still unbiased, but fails to be BLUE as it fails to give the most efficient parameter estimates. In this scenario, a strategy which transforms the regressions variables so that the error has no structure is in order. In time-series analysis, the problem of autocorrelation between the residual values is a common one.  There are several ways to approach the transformations necessary to ensure BLUE estimates, and the previous post used the following method to gain asymptotic efficiency and improve our estimates:

1. Estimate the OLS regression

2. Fit OLS residual to an AR(p) process using the Yule-Walker Method and find the value of p.

3.  Re-estimate model using Generalized Least Squares fit by Maximum Likelihood estimation, using the  estimated p from 2, as the order for your correlation residual term.

4. Fit the GLS estimated residuals to an AR(p) process and use the estimated p's as the final parameter estimates for the error term.  

What have we done?  First we have to find out what the error term autocorrelation process is. What order is p? In order to find this out we fit the OLS residuals to an AR(p) using the Yule-Walker method. Then we take the order p of our estimated error term and run a GLS regression with an AR(p) error term.  This will give us better estimates for our model.  Research has shown that GLS estimators are asymptotically more efficient than OLS estimates almost one-hundred percent of the time. If you notice in every single regression, the GLS estimator with a twice iterated AR(p) error terms consistently results in a lower standard deviation of the residual value. Therefore the model has gained efficiency which translates into improved confidence intervals.  Additionally, by fitting the GLS residuals to an AR(p) we remove any autocorrelation(or structure) that may have been present in the residual.  

Testing For Model Miss-specification and Omitted Variable Bias

The Ramsey RESET test (Regression Specification Error Test) is designed to detect omitted variable bias and incorrect functional form. Rejection of H_{0} implies that the original model is inadequate and can be improved.  A failure to reject H_{0} conveys that the test has not been able to detect any miss-specification.

Unfortunately our models of short-term risk premia over both estimation periods reject the null hypothesis, and thus suggest that a better model is out there somewhere.  Correcting for this functional miss-specification or omitted variable bias will not be pursued here, but we must keep in mind that our model can be improved upon and is thus not BLUE.  

In R you can run the Ramsey Reset test for standard lm functions using the library lmtest:

>library(lmtest)

> resettest(srp1.lm)

RESET test

data:  srp1.lm 

RESET = 9.7397, df1 = 2, df2 = 91, p-value = 0.0001469

For GLS objects however you'll need to do it manually and that procedure will not be outline here.  Although if you really want to know please feel free to email or leave a comment below.  

Addressing Multicollinearity

In the original formulation of the model there existed an independent variable called CreditMarketSupport, that was very similar to our FedBalance variable.  Both variables are percentages and shared the same numerator while also having very similar denominators.  As a result we had suffered from a condition called exact collinearity as the correlation between these two variables was nearly one.

> cor(FedBalance1,CreditMarketSupport1)

0.9994248

With exact collinearity we were unable to obtain a least squares estimate of our ß coefficients and these variables were behaving opposite of what we were expecting.  This violated one of our least squares assumptions SR5 which states that values of x_{ik} are not exact linear functions of the other explanatory variables.  To remedy this problem, we removed CreditMarketSupport from the models and we are able to achieve BLUE estimates.

 Suspected Endogeniety

In our estimation of long-term risk premia over the first time period we suspect endogeniety in the cyclical variable Output Gap.  In order to remedy this situation we replace it with an instrumental variable - the percentage change in S&P 500 and perform the Hausman Test which is laid out as follows:

H_{0}: delta = 0 (no correlation between x_{i} and µ_{i})

H_{1}: delta ≠ 0 (correlation between x_{i} and µ_{i})

When we perform the Hausman Test using S&P 500 as our instrumental variable our delta ≠ 0 and is statistically significant.  This means that our Output Gap variable is indeed endogenous and correlated with the residual term.  If you want to learn more about the Hausman Test and how to perform it in R please leave a comment or email me and i'll make sure to get the code over to you.  When we perform the Two Stage Least Squares Regression to correct for this not a single term is significant.  This can be reasonably be attributed to the problem of weak instruments.  The 2 Stage Least Squares Estimation is provided below. Since, the percentage change in the S&P500 was only correlated with the Output Gap 0.110954, there is strong reason to suspect that weak instruments are the source of the problem.  We will choose to not locate a proper instrumental variable to emulate the Output Gap, instead we will keep in mind that we have an endogenous variable when interpreting our coefficient estimates which will now end up being slightly biased. 

Below is how to perform a two-stage least squares regression in R when your replacing an endogenous variable with an exogenous one. First you'll need to load the library sem into R. In the below regression the first part includes all the variables from the original model and the second part lists all of our exogenous and instrumental variables which in this case is just the percentage change in the S&P 500.

> tSLRP1<-tsls(lrp1~yc1+CP1+FF1+default1+Support1+ER1+FedGDP1+FedBalance1+govcredit1+ForeignDebt1+UGAP1+OGAP1,~ yc1+CP1+FF1+default1+Support1+ER1+FedGDP1+FedBalance1+govcredit1+ForeignDebt1+sp500ch+OGAP1 )

> summary(tSLRP1)

 2SLS Estimates

Model Formula: lrp1 ~ yc1 + CP1 + FF1 + default1 + Support1 + ER1 + FedGDP1 + 

    FedBalance1 + govcredit1 + ForeignDebt1 + UGAP1 + OGAP1

Instruments: ~yc1 + CP1 + FF1 + default1 + Support1 + ER1 + FedGDP1 + FedBalance1 + 

    govcredit1 + ForeignDebt1 + sp500ch + OGAP1

Residuals:

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 

 -9.030  -1.870   0.021   0.000   2.230   7.310 

             Estimate Std. Error  t value Pr(>|t|)

(Intercept)  -5.28137   44.06906 -0.11984   0.9049

yc1          -1.48564   10.60827 -0.14005   0.8889

CP1          -0.01584    0.09206 -0.17204   0.8638

FF1           0.20998    2.43849  0.08611   0.9316

default1     -7.16622   65.35728 -0.10965   0.9129

Support1      6.39893   47.72244  0.13409   0.8936

ER1           4.56290   35.91837  0.12704   0.8992

FedGDP1       1.86392    9.16081  0.20347   0.8392

FedBalance1   0.73087   12.96474  0.05637   0.9552

govcredit1    0.17051    0.89452  0.19062   0.8492

ForeignDebt1 -0.22396    1.41749 -0.15799   0.8748

UGAP1         4.55897   35.33446  0.12902   0.8976

OGAP1         0.01331    0.09347  0.14235   0.8871

Residual standard error: 3.3664 on 93 degrees of freedom

Notice that our model now doesn't have any significant terms.  This is why we will choose to ignore the endogeniety of our Output Gap and probably Unemployment Gap variables.  Correcting for endogeniety does more harm than good in this case.

Results and Concluding Thoughts

As this paper hopefully shows, the Feds actions did directly impact the easing of broader credit conditions in the financial markets.  

Over our first estimation period from 1971 to 1997 we find that the Fed's support of Depository Institutions as a percentage of savings and time deposits is positively related to the short-term risk premia. Specifically we find that a 1 percentage point increase in Support leads to a 2.1 percent increase in short-term risk premia. This was as expected because Depository Institutions would only borrow from the Fed if no other options existed. We also find that a 1 percentage point increase in the federal funds rate leads to a .19 percentage increase in short-term risk premia.  This is consistent with our original hypothesis as an increased FF puts positive pressure on short-term rates like the 3 month commercial paper rate, thus resulting in an widened spread.  With respect to long-term risk premia, we find that a 1 percentage point increase in FF leads the long-term risk premia to decrease by .66 percentage points and a 1 percent increase in the federal funds rate leads to a .07 decrease in the long-term risk premia.

Over our second estimation period the composition of the Feds balance sheet is considered.  We see that the CCLF  did decrease short-term risk premiums, with every one percent increase translating to a decrease in short-term risk premia by .1145 percentage points.  Another important result is that Fed purchases of Agency Debt and Agency MBS did have a significant, although almost negligible effect on short-term risk premia.  One surprising result with the estimation of the long-term risk premia is that our Fed balance sheet size variable has a sign that is opposite of what we expected and its significance is particularly surprising.  This may be expected since this period is largely characterized by both a shrinking balance sheet and narrowing risk premia as investments were considered relatively safe.  However towards the end of the period risk premiums shot up and only after did the size of the balance sheet also increase, thus the sample period may place too much weight towards the beginning of the time period and not enough towards the end. This is a reasonable assumption given that our estimate of the balance sheet size showed a large negative impact on risk premia over our longer estimation period.  

A complete version of the paper can be located here.

Please people keep dancing and we'll delve further into some additional econometrics tests next week. 

Monetary Policy and Credit Easing pt. 6: Empirical Estimation and Methodology

IT is now appropriate to lay out our two regression models in full for empirical estimation over our two separate time periods. The first estimation is from 4/1/71 to 7/1/97 and the second is from 4/1/01 to 4/1/11. The methodology employed in the estimation of these two models is a procedure using Generalized Least Squares with a Cochrane-Orcutt, style iterated residual value.  For those that wish to perform the same regressions at home I have provided the following links to my data.  This is for the estimation period from 1971 to 1997 and this one is for the 2001 to 2011 estimations. The following four steps were taken for each estimation:

1. Estimate the OLS regression

2. Fit OLS residual to an AR(p) process using the Yule-Walker Method and find the value of p.

3.  Re-estimate model using Generalized Least Squares fit by Maximum Likelihood estimation, using the  estimated p from 2, as the order for your correlation residual term.

4. Fit the GLS estimated residuals to an AR(p) process using the Yule-Walker Method and use the estimated p's as the final parameter estimates for the error term.

The end goal of the above procedure is to have our models be asymptotically BLUE or the Best Linear Unbiased Estimators.  The implies that they have the smallest variance out of all the models that meet the rest of the Gauss-Markov assumptions.  A little background behind the methodology is in order.  First we will perform the standard Ordinary Least Squares (OLS) regression on our dependent variables.  This will give us at the very least unbiased estimators.  Second we take the residuals from the first step and fit them to an AR(p) process as selected by the Yule-Walker Method.  This method selects the optimal lag that characterizes the autocorrelation in the residuals.  We automatically take this step, due to the well known fact that most time-series suffer from autocorrelation problems as verified by our correlograms.  Then we re-estimate the regression using Generalized Least Squares which adjusts each term and divides them by their individual variances, while also incorporating the AR(p) lag that we discovered during the previous step.  The final step, which fits our GLS models residuals to an AR(p) process is what leads to our asymptoticly efficient results. We lay out our first estimation period models below.

First Estimation:  4/1/71 to 7/1/97 

1. Monetary Policies Impact On Short-term Risk Premiums

Our first model which seeks to answer how monetary policy impacts the risk premia on short-term commercial paper as estimated over our first time period 4/1/71 to 7/1/97 is as follows:

SR^{premium}_{t}=ß_{0}+ß_{1}*FedBalance^{size}_{t}+ß_{2}*Support_{t} + ß_{3}*UGAP_{t}+ ß_{4}*FF_{t}+ ß_{5}*ER_{t}+ ß_{6}*YC_{t}+ ß_{7}*Default^{spread}_{t}+ ß_{8}*CP_{t}+ ß_{9}*OGAP_{t} + ß_{10}*FedGDP_{t}+ ß_{11}*govcredit_{t}+ ß_{12}*ForeignDebt_{t} +  µ_{t}

Where,

SR^{premium}_{t} = Short-term Risk Premium at time, t

Support_{t}= Fed's funds at depository institutions as a percentage of their main financing streams at time, t

FedBalance^{size}_{t}=  The Fed's credit market asset holdings as percentage of the total credit market assets at time, t

FF_{t}= Federal Funds rate at time, t

ER_{t}= Excess Reserves of Depository Institutions at time, t

YC_{t}= Yield curve at time, t

Default^{spread}_{t}= Default Spread between BAA_{t} & AAA_{t} rated bonds at time, t

CP_{t} = Corporate Profits After Tax at time, t

FedGDP_{t}= Fed's holdings of total public debt as a percentage of GDP at time, t

govcredit_{t}= Government Holdings Of Domestic Credit Market Debt As A Percentage Of The Total at time, t

ForeignDebt_{t}= Foreign Holdings of Federal Debt As A Percentage Of The Total at time, t

UGAP_{t} = Unemployment gap at time, t

OGAP_{t} = Output gap at time, t

µ_{t}= error term at time, t

R DATA WORK

So now it is time for the long awaited econometrics work in R. First thing you'll want to do is read the data into R from your data file which is in this case is the Earlreg file.

> earl<- read.csv("/Users/stevensabol/Desktop/R/earlreg.csv",header = TRUE, sep = ",")

Then you define your variable names so you can easily manipulate your data in R. So when you open the official .csv data file take a look at the variable names and rename them using the following procedure.  

>yc1<-earl[,"yc"]

After you define what you call everything you're then free to go crazy and run regressions.  Below is how you run the standard Ordinary Least Squares regression.  The lm function enables you to run linear regressions:

1. Estimate the OLS regression

>srp1.lm=lm(srp1~yc1+CP1+FF1+default1+Support1+ER1+FedGDP1+FedBalance1+govcredit1+ForeignDebt1+UGAP1+OGAP1)

In order to get the output you have to use the summary function:

> summary(srp1.lm)

Call:

lm(formula = srp1 ~ yc1 + CP1 + FF1 + default1 + Support1 + ER1 + 

    FedGDP1 + FedBalance1 + govcredit1 + ForeignDebt1 + UGAP1 + 

    OGAP1)

Residuals:

     Min       1Q   Median       3Q      Max 

-1.04289 -0.20145 -0.04041  0.15230  1.21044 

Coefficients:

               Estimate Std. Error t value Pr(>|t|)    

(Intercept)  -2.7591194  1.0359966  -2.663  0.00912 ** 

yc1           0.1320996  0.0580500   2.276  0.02516 *  

CP1          -0.0022773  0.0073773  -0.309  0.75825    

FF1           0.1699788  0.0340654   4.990 2.81e-06 ***

default1      0.4382965  0.1876685   2.335  0.02167 *  

Support1      2.2383850  0.6660140   3.361  0.00113 ** 

ER1           0.3351508  0.3017644   1.111  0.26959    

FedGDP1       0.3031938  0.2558144   1.185  0.23895    

FedBalance1   0.4014920  0.3477547   1.155  0.25124    

govcredit1   -0.0928817  0.0401603  -2.313  0.02294 *  

ForeignDebt1 -0.0068900  0.0215393  -0.320  0.74977    

UGAP1        -0.0912273  0.0520491  -1.753  0.08295 .  

OGAP1         0.0006669  0.0014895   0.448  0.65536    

---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0.3789 on 93 degrees of freedom

Multiple R-squared: 0.6474, Adjusted R-squared: 0.6019 

F-statistic: 14.23 on 12 and 93 DF,  p-value: 2.642e-16 

Next we perform step 2 which is:

2. Fit OLS residual to an AR(p) process using the Yule-Walker Method and find the value of p.

> srp.lmfit<-ar.yw(srp1.lm$res)

> srp.lmfit

Call:

ar.yw.default(x = srp1.lm$res)

Coefficients:

     1  

0.2535  

Order selected 1  sigma^2 estimated as  0.1201 

So the Yule-Walker methodology fits the residual series to an AR(1) process.

3.  Re-estimate model using Generalized Least Squares fit by Maximum Likelihood estimation, using the  estimated p from 2, as the order for your correlation residual term.

In order to run a GLS regression your going to need to load the nlme package:

> library(nlme)

Then you can go crazy:

>srp1.gls=gls(srp1~yc1+CP1+FF1+default1+Support1+ER1+FedGDP1+FedBalance1+govcredit1+ForeignDebt1+UGAP1+OGAP1, corr=corARMA(p=1,q=0),method="ML")

> summary(srp1.gls)

The following output is produced:

Generalized least squares fit by maximum likelihood

  Model: srp1 ~ yc1 + CP1 + FF1 + default1 + Support1 + ER1 + FedGDP1 + FedBalance1 + govcredit1 + ForeignDebt1 + UGAP1 + OGAP1 

  Data: NULL 

       AIC      BIC    logLik

  100.7905 140.7421 -35.39526

Correlation Structure: AR(1)

 Formula: ~1 

 Parameter estimate(s):

      Phi 

0.3696665 

Coefficients:

                  Value Std.Error   t-value p-value

(Intercept)  -3.0219486 1.2595942 -2.399145  0.0184

yc1           0.1929605 0.0640627  3.012054  0.0033

CP1          -0.0060642 0.0071791 -0.844700  0.4004

FF1           0.1918066 0.0362894  5.285466  0.0000

default1      0.5292204 0.2060591  2.568293  0.0118

Support1      2.1086204 0.7405128  2.847514  0.0054

ER1           0.5651430 0.2770125  2.040135  0.0442

FedGDP1       0.1028773 0.3143122  0.327309  0.7442

FedBalance1   0.7845392 0.4130914  1.899190  0.0606

govcredit1   -0.1240196 0.0524191 -2.365922  0.0201

ForeignDebt1  0.0009822 0.0278623  0.035252  0.9720

UGAP1        -0.1266050 0.0657633 -1.925161  0.0573

OGAP1        -0.0014094 0.0014328 -0.983623  0.3279

 Correlation: 

             (Intr) yc1    CP1    FF1    deflt1 Spprt1 ER1    FdGDP1 FdBln1 gvcrd1

yc1          -0.267                                                               

CP1           0.054 -0.062                                                        

FF1          -0.308  0.726 -0.012                                                 

default1      0.081 -0.208  0.235 -0.342                                          

Support1     -0.005 -0.109 -0.107 -0.419  0.137                                   

ER1          -0.208  0.077 -0.081  0.067 -0.180  0.028                            

FedGDP1      -0.728 -0.059 -0.048  0.083 -0.057 -0.002 -0.308                     

FedBalance1   0.461  0.250  0.020  0.208  0.036 -0.081  0.445 -0.887              

govcredit1   -0.570 -0.233 -0.095 -0.291 -0.261  0.072  0.068  0.666 -0.784       

ForeignDebt1 -0.475  0.132  0.006 -0.092  0.093  0.219  0.057  0.059 -0.045  0.227

UGAP1        -0.048 -0.193 -0.062  0.085 -0.447  0.150  0.090  0.045  0.064 -0.053

OGAP1        -0.029  0.092  0.295  0.062 -0.208 -0.024  0.053 -0.021  0.013  0.056

             FrgnD1 UGAP1 

yc1                       

CP1                       

FF1                       

default1                  

Support1                  

ER1                       

FedGDP1                   

FedBalance1               

govcredit1                

ForeignDebt1              

UGAP1        -0.016       

OGAP1         0.064  0.041

Standardized residuals:

        Min          Q1         Med          Q3         Max 

-3.08026826 -0.62589269 -0.08409222  0.39781537  3.24233325 

Residual standard error: 0.3634024 

Degrees of freedom: 106 total; 93 residual

After you perform this step you have to refit the residuals in order to get serially uncorrelated terms.  

4. Fit the GLS estimated residuals to an AR(p) process using the Yule-Walker Method and use the estimated p's as the final parameter estimates for the error term.  

> s1glsres.ar<-ar.yw(srp1.gls$res)

> s1glsres.ar

Call:

ar.yw.default(x = srp1.gls$res)

Coefficients:

     1  

0.3718  

Order selected 1  sigma^2 estimated as  0.1163 

In order to see the results of these actions please refer to image below:

The Ljung-box Q is a test for autocorrelation between the lags in the error terms. Ideally in order to meet the BLUE criteria we have to reject the the null hypothesis for autocorrelation at each lag.  We can see that both our original OLS  and GLS estimations fail to pass the Ljung-Box Q. However when we readjust the error terms the final time we get residuals that are serially uncorrelated with each other.  

In order to get the above graph you have to first load the package that will allow you to perform the Ljung-Box Q plot:

> library(FitAR)

Then you can proceed from there and define how many plots should be in one picture.  In the above image we have 9 therefore:

> par(mfrow=c(3,3))

Then you can start adding in your plots.  Below is the code for producing the plots for the fitted GLS residuals.

> acf(s1glsres.ar$res[-(1)])

> pacf(s1glsres.ar$res[-(1)])

> LBQPlot(s1glsres.ar$res[-(1)])

We include the [-(1)] because exclude the first observation since we have an AR(1) process.

The same steps above can be applied to any time-series regression model.  In the next post we will discuss how to get some summary statistics.  Please keep dancin'

Steven J.

Monetary Policy & Credit Easing pt. 5: Explanatory Variables Continued...

Capturing Treasury Supply Effects

WE will need to account for things other than the Fed that influenced risk premia as they relate to Treasury supply. The following three variables are meant to accomplish such a thing:

1. Federal Reserves holdings of total public debt as a percentage of GDP

2. Total government holdings of domestic credit market debt as a percentage of the total

3. Foreign holdings of government debt as a percentage of total public debt

1. Fed's holdings of total public debt as a percentage of GDP

Federal Reserve holdings of total public debt as a percentage of GDP is important because it controls for how much Federal Government support the Fed is accounting for.  It is especially pertinent to our second estimation as the Feds holdings of total public debt relative to GDP increased sharply.  Operationally, we define this variable as:

FedGDP_{t}= {GovDebt_{t}^{Fed}\ GDP_{t}}x 100

Where,

GovDebt_{t}^{Fed}=Federal Debt Held by Federal Reserve Banks (FDHBFRBN) at time, t

GDP_{t} = Gross Domestic Product, 1 Decimal (GDP) at time, t

We expect that this variable will move in line with both short-term and long-term risk premiums. Therefore:

H_{0}: ß ≤ 0 vs. H_{a}: ß > 0

Data Issues

The time-series necessary for this variable is provided by FRED and the details are as listed:

(a) Federal Debt Held by Federal Reserve Banks (FDHBFRBN), Quarterly, End of Period, Not Seasonally Adjusted, 1970-01-01 to 2011-0
(b) Gross Domestic Product, 1 Decimal (GDP), Quarterly, Seasonally Adjusted Annual Rate, 1947-01-01 to 2011-07-01

2. Government Holdings Of Domestic Credit Market Debt As A Percentage Of The Total

It would be wise to include a variable that account for fiscal policies support of the financial markets.  This we can define as Federal Government holdings of credit market assets as a percentage of the total outstanding. To account for total government support of the financial markets we will use the following variable: govcredit.

govcredit_{t}={ CAssets_{t}^{Gov}\ CAssets_{t}^{Total} }x 100

Where,

CAssets_{t}^{Gov} = Total Credit Market Assets Held by Domestic Nonfiancial Sectors - Federal Government (FGTCMAHDNS) at time, t

CAssets_{t}^{Total} = Total Credit Market Assets Held by Domestic Nonfiancial Sectors (TCMAHDNS) at time, $t$

We expect that this variable will reduce both short-term and long-term risk premiums. Therefore:

H0: ß ≥ 0 vs. Ha: ß < 0

Data Issues

The time-series necessary for this variable is provided by FRED and the details are as listed:

(a) Total Credit Market Assets Held by Domestic Nonfiancial Sectors - Federal Government (FGTCMAHDNS), Quarterly, End of Period, Not Seasonally Adjusted, 1949-10-01 to 2011-04-01
(b) Total Credit Market Assets Held by Domestic Nonfiancial Sectors (TCMAHDNS), Quarterly, End of Period, Not Seasonally Adjusted, 1949-10-01 to 2011-04-01

3. Foreign Holdings of Federal Debt As A Percentage Of The Total

This variable labeled ForeignDebt_{t} seeks to capture the impact that foreign holdings of United States government debt have on both short-term and long-term risk premia.  Theory would suggest that as foreign holdings go up risk-premia would go down. Operationally this variable is defined as follows:

ForeignDebt_{t} = {GovDebt_{t}^{Foreign}\ total public debt_{t}} x 100

Where,

GovDebt_{t}^{Foreign}= Federal Debt Held by Foreign & International Investors (FDHBFIN) at time, t

total public debt_{t}= Federal Government Debt: Total Public Debt (GFDEBTN) at time, t

Our ß coefficient on this variable is expected to be negative for both short-term and long-term risk premia and therefore:

H_{0}: ß ≥ 0 vs. H_{a}: ß < 0


Data Issues

The following data comes from FRED and the details are as follows:

(a) Federal Debt Held by Foreign & International Investors (FDHBFIN), Quarterly, End of Period, Not Seasonally Adjusted, 1970-01-01 to 2011-04-01
(b) Federal Government Debt: Total Public Debt (GFDEBTN), Quarterly, End of Period, Not Seasonally Adjusted, 1966-01-01 to 2011-04-01

Accounting For Cyclicality

We include two variables to help account for cyclicality in the overall economy.  Both are relevant as the Fed uses these variables in its decision making process.  For example in setting the federal funds rate, the Fed is said to have used a Taylor Rule that incorporated both the output gap and unemployment gap in its objective function.  Thus incorporating these variables may present a problem of endogeneity over a short-part of our sample(like when a taylor-rule was said to be used), but these effects we will choose to ignore.  The two cyclical variables we shall use are the output gap and the unemployment gap. The output gap is defined,

OGAP_{t}= Potential GDP_{t} – GDP_{t} at time, t

Where,

Potential GDP_{t}=Nominal Potential Gross Domestic Product (NGDPPOT) at time, t

GDP_{t}=Gross Domestic Product, 1 Decimal (GDP) at time, t

Our unemployment gap is defined in a similar fashion:

UGAP_{t}= NROU_{t} – UNRATE_{t} at time, t

Where,

NROU_{t}= Natural Rate of Unemployment (NROU) at time, t

UNRATE_{t}= Civilian Unemployment Rate (UNRATE) at time, t

Theoretically we assume that over the long-run as both of these variables increase the long-term risk premia increase.  Over the short-run regressions we would expect these variables to have almost no significant effect as that time period is cluttered with many short-term things impacting risk-premia. Additionally for the short-term risk premia we would expect either a negative relationship or no relationship.  This is because many things that impact the long-term risk premia one way have an opposite sign with respect to the short-term risk premia.

Data Issues
The data for these cyclical variables is provided by FRED and their details are laid out as follows:

(a) Civilian Unemployment Rate (UNRATE), Monthly, Seasonally Adjusted, 1948-01-01 to 2011-10-01 
(b) Natural Rate of Unemployment (NROU), Quarterly, 1949-01-01 to 2021-10-01 
(c) Nominal Potential Gross Domestic Product (NGDPPOT), Quarterly, 1949-01-01 to 2021-10-01 
(d) Gross Domestic Product, 1 Decimal (GDP), Quarterly, Seasonally Adjusted Annual Rate, 1947-01-01 to 2011- 07-01

The next post gets into the R analysis and lays out our model in full.

Monetary Policy & Credit Easing pt. 4: More Independent Variable Definitions

Support for Depositary Institutions

This variable will account for the Federal Reserves support of Depository Institutions through direct lending to these institutions.  Support will be measured by how much the Fed made up for any shortfalls in Depository Institutions main source of cash- time and savings deposits. Federal Reserve support for our first estimation period is operationalized as follows:

Support_{t}={TotalBorrowingFed^{DI}_{t}\ Total Time & Savings Deposits_{t}^{DI}}x 100

Where,
Support_{t}= Fed funds at depository institutions as a percentage of their main financing streams (total savings and time deposits) at time, t

TotalBorrowingFed^{DI}_{t}= Total Borrowings of Depository Institutions from the Federal Reserve (BORROW) at time, t

Total Time & Savings Deposits_{t}^{DI}= Total Time and Savings Deposits at All Depository Institutions (TOTTDP) at time, t

For our second estimation period from 4/1/01 to 4/1/11 we will use a different variable that excludes time-deposits as the series that we would ideally like to use above was discontinued in 2006.

Support_{t}={TotalBorrowingFed^{DI}_{t}\ Total Savings Deposits_{t}^{DI}}x 100

Where,
Support_{t}= Fed funds at depository institutions as a percentage of their main financing stream (total saving deposits) at time, t

TotalBorrowingFed^{DI}_{t}= Total Borrowings of Depository Institutions from the Federal Reserve (BORROW) at time, t

Total Savings Deposits_{t}^{DI}= Total Savings Deposits at all Depository Institutions (WSAVNS) at time, t

The expected beta coefficient should be positively related to short-term risk premia, as tighter credit conditions require Depository Institutions to go to the Fed for help.  Only after the risk-premia goes up and these institutions have no where else to go do they borrow from the Fed at the discount rate.

We expect the effect of lending support to depository institutions to be positively related to short-term risk premia, therefore:

H_{0}: ß ≤ 0 vs. H_{a}: ß > 0

Furthermore, we expect Fed support to depository institutions to have a negative effect on long-term risk premia because of the expectations component.  As the Fed steps in with lending support, markets calm their fears about the future.  This is directly the opposite of short-term risk premia as the support in that situation is in direct response to the risk premia. Therefore,

H_{0}: ß ≥ 0 vs. H_{a}: ß < 0

Data Issues

The following time-series data were provided by FRED and the details are as follows:

(a) Total Borrowings of Depository Institutions from the Federal Reserve (BORROW), Monthly, Not Seasonally Adjusted, 1919-01-01 to 2011-10-01


(b) Total Savings Deposits at all Depository Institutions (WSAVNS), Weekly, Ending Monday, Not Seasonally Adjusted, 1980-11-03 to 2011-10-17


(c) Total Time and Savings Deposits at All Depository Institutions (DISCONTINUED SERIES) (TOTTDP), Monthly, Seasonally Adjusted, 1959-01-01 to 2006-02-01

The Federal Funds Rate

The motivation behind putting the Federal Funds rate into our regression model is simple.  This is the main policy tool that the Fed has used to manipulate short-term credit conditions and influence the rate of inflation.  The Fed controls this rate in hopes of influencing other rates such as the Prime Bank Loan Rate along with other short term credit instruments like Commercial Paper. Additionally, this variable is very easy to account for because it requires virtually zero manipulation.  Notationally we will define it in the following manner:

FF_{t}= Federal Funds rate at time, t

The expected beta coefficient should be positively related to the short-term risk premia and negatively related to long-term risk premia.  They theory is that by lowering the federal funds rate, or the rate at which banks lend to each other, the Fed is encouraging banks to lend and thus ease credit conditions.  When the Fed feels that tightening is appropriate, maybe the result of a jump in inflation expectations or an deep acceleration in the economy, they respond by raising the federal funds rate.  This tightens credit conditions and thus theoretically at least, should result in an increase in short-term risk premia. The opposite holds true for the federal funds rate effect on long-term risk premia.  Since long-term rates are a function of shorter-term rates, investors are inclined to sell Treasuries which decreases the spread between Aaa and 10 year nominal treasuries, therefore decreasing long-term risk premia.

For short-term risk premiums we expect the federal funds rate to be positively related:

H_{0}: ß ≤ 0 vs. H_{a}: ß > 0

For long-term risk premiums we expect the federal funds rate to be negativity related:

H_{0}: ß ≥ 0 vs. H_{a}: ß < 0

Data Issues

We get the series for the federal funds rate from FRED and the details are as follows:

(a) Effective Federal Funds Rate (FF), Weekly, Ending Wednesday, 1954-07-07 to 2011-10-19

Interest Paid On Excess Reserves

This variable is also easy to account for because the introduction of interest paid on reserves has a direct impact on the physical quantity of excess reserves of depository institutions held at the Federal Reserve.  The motivation behind this variable is that banking institutions would not hold excess reserves with the Fed without some compensation i.e. that opportunity cost has to be greater than zero.  That is where the interest paid on excess reserves come in.  Without it there is an opportunity cost of letting the reserves sit with the fed instead of seeking more profitable safe havens for their cash. That is why we will be using the quantity of excess reserves as our explanatory variable and notationally defining it as follows:

ER_{t}= Excess Reserves of Depository Institutions at time, t

When the Fed initiated its policy of paying interest on reserves it created an incentive for banks to shore up their finances with the Fed.  It gave the Fed a way to conduct large scale asset purchases without suffering inflationary consequences. As long as the reserves are held with the Fed, they cannot be inflationary therefore an increase in excess reserves at the Fed is contractionary. Additionally, since the Fed had not initiated the policy of paying interest on reserves until October of 2008 there was no incentive for banks to hold any excess reserve balances before then. That is why our two estimation periods must have two different hypothesis tests for this variable:

For our estimation covering the 4/1/71 to 7/1/97 time period the interest paid on excess reserves policy was non-existent and therefore:

H_{0}: ß = 0 vs. H_{a}: ß ≠ 0

In other words, we are looking to not reject the null hypothesis that beta is equal to zero.

For our second estimation covering the 4/1/01 to 4/1/11 time period the interest paid on excess reserves policy was in effect, if only for a short-time before the end of the sample and therefore:

H_{0}: ß ≤ 0 vs. H_{a}: ß > 0

Data Issues

We get the series from FRED and the details are as follows:

(a) Excess Reserves of Depository Institutions (EXCRESNS), Monthly, Not Seasonally Adjusted, 1959-01-01 to 2011-09-01

Control Variables: Accounting For Factors Outside of Monetary Policy

We must account for changes in the risk premia that aren't necessarily related to monetary policy.  These include things like market fear, corporate default risk and controlling for changes due to underlying fundamentals like Corporate Profits After Tax.

The Yield Curve

The motivation behind including the yield curve is due to its known predictive power of economic growth and recessionary risk. As recessionary risk increase investors are more likely to put their funds into the safest assets with the highest return.  Historically, this involves the purchase of longer-term Treasuries as they have zero default risk.  When a bond is purchased its price goes up, and its effective yield decreases so the slope of the yield curve or the spread between the most liquid and shortest maturity bond and the most liquid, longest maturity bond decreases or flattens. As this slope flattens we expect risk premia to increase for longer maturity and less liquid debt instruments. As the yield curve flattens we expect T-bills to be sold and longer-term Treasuries to be purchased.  This puts upward pressure on T-bill rates, thus narrowing the spread between Commercial Paper and T-bills and reducing the short-term risk premia.
The yield curve as we define it is the spread between the 10-Year Nominal Treasury Note rate and the 3-Month Nominal Secondary Market Treasury Bill rate.  Notationally,

YC_{t}=GS10_{t} – TB3MS_{t}

Where,

YC_{t}= Yield curve at time, t

GS10_{t}= 10-Year Treasury Constant Maturity Rate at time, t

TB3MS_{t}= 3-Month Treasury Bill: Secondary Market Rate at time, t

The beta coefficient for both of our estimation periods is expected to be negatively related to long-term risk premia. Therefore:

H_{0}: ß ≥ 0 vs. H_{a}: ß ≤ 0

The beta coefficient for both of our estimation periods is expected to be positively related to short-term risk premia. Therefore:

H_{0}: ß ≤ 0 vs. H_{a}: ß > 0

Data Issues

We get the two times-series data from FRED and the details are as follows:

(a) 10-Year Treasury Constant Maturity Rate (GS10), Monthly, 1953-04-01 to 2011-09-01


(b) 3-Month Treasury Bill: Secondary Market Rate (TB3MS), Monthly, 1934-01-01 to 2011-09-01

Stock Market Volatility

We will control for market fear by including a measure of stock market volatility into our regression model. We define volatility as follows:

 Volatility_{t}= CBOE DJIA Volatility Index (VXDCLS) at time, t

The motivation behind this control variable is that the returns from owning stocks become more volatile in times of fear. Thus the risk premia on assets that aren't risk free like corporate bonds may increase in response to this market volatility.

This variables is literally only relevant in the regression on long-term premiums for our second estimation period. We cannot reasonably assume it has any effect on short-term risk premia because stock market volatility signals the fear of financial markets which leads to flight to safety into longer term treasuries not short-term commercial paper. The reason is that an investor probably couldn't even have the capital to buy commercial paper to begin with and secondly when fear strikes investors tend to pour their capital into longer term treasuries because they can pick up some extra yield. This would widen the spread between Aaa and Treasuries thus increasing the risk premium.  Therefore we include this variable only in our second estimation and its only real effect will be on the long-term risk premia.

For the regression over our second estimation period, increased volatility is expected to increase the long-term risk premium:

H_{0}: ß ≤ 0 vs. H_{a}: ß > 0

For the regression over our second estimation period, volatility is not expected to have an effect on short-term risk premia, therefore:

H_{0}: ß = 0 vs. H_{a}: ß ≠ 0

In other words, we are looking to not reject the null hypothesis that $\beta$ is equal to zero.

Data Issues

We get the times-series data from FRED and the details are as follows:

(a) CBOE DJIA Volatility Index (VXDCLS), Daily, Close, 1997-10-07 to 2011-11-02

Corporate Bond Default Risk

It would be prudent to control for the perceived credit default risk of the corporate bonds market. To do this we use the spread between the AAA and BAA rates bonds which would theoretically correspond to increased compensation for the risk of a default. The reason being we want to see how much the Fed's actions influence the risk premia and factor out movements in the spread that may incorporate other things like flat out default risk. Ideally we would like to use a Credit Default Swap Index to control for default risk as it would help us control directly for default risk and not other things like liquidity risk, but given the sample length data limitations we are forced to stick with what we've got.

The control variable we use for corporate default risk is the spread between Moody's rated Baa's and Aaa's. This is used because data exists for the full length of our desired samples and therefore is operationalized as follows:

Default^{spread}_{t}=BAA_{t} – AAA_{t}

Where,

BAA_{t}= Moody's Seasoned Baa Corporate Bond Yield at time, t

AAA_{t}= Moody's Seasoned Aaa Corporate Bond Yield at time, t

The beta coefficient on our corporate default control variable is expected to be positive in our regression on long-term rates since an increase in the spread between BAA and AAA would indicate that these bonds expected default risk would increase:

H_{0}: ß ≤ 0 vs. H_{a}: ß > 0

In our regressions on short-term risk premia, the expected effect of this control variable is effectively zero as this variable deals with longer-term interest rates not exactly pertinent to short-term financing like commercial paper or Treasury Bills:

H_{0}: ß = 0 vs. H_{a}: ß ≠ 0

In other words, we are looking to not reject the null hypothesis that $\beta$ is equal to zero.

Data Issues

For the AAA and BAA data we use FRED:

(a) Moody's Seasoned Aaa Corporate Bond Yield (AAA), Monthly, 1919-01-01 to 2011-09-01


(b) Moody's Seasoned Baa Corporate Bond Yield (BAA), Monthly, 1919-01-01 to 2011-09-01

Corporate Profits After Tax

As corporate profits after tax increase the risk premium on corporate bonds decrease.  This fundamentally negative relationship should be controlled for in our regression model.  Operationally, this is defined as:

CP_{t} = Corporate Profits After Tax at time, t

The beta coefficient on our CP control variable should be negatively related to our dependent variables.  This is because as corporate profits after tax increase the risk that they will renege on their debt obligations will decrease. This gives us the following test.

H_{0}: ß ≥ 0 vs. H_{a}: ß < 0

Data Issues

We get the times-series data from FRED and the details are as follows:

(a) Corporate Profits After Tax (CP), Quarterly, Seasonally Adjusted Annual Rate, 1947-01-01 to 2011-04-01

Keep dancin'

Steven J.

Monetary Policy & Credit Easing pt. 3: Accounting For The Composition of The Fed's Balance Sheet & Credit Easing

Credit Easing shifts the composition of the balance sheet away from default-free assets towards assets with credit risk. An example of Credit Easing which is pertinent to our testing the effects of monetary policy on commercial paper is the Commercial Paper Funding Facility. Implementation of this facility involved the U.S. central bank selling T-bills and purchasing commercial paper of similar maturity.  This shift in composition leaves the size and average maturity of bank assets on the Fed's balance sheet unchanged. When the Fed purchases an asset like commercial paper, it lowers the supply of this asset to private investors. This scarcity has the effect of boosting its price and pushing down its yield. In the absence of private demand for the risky asset, the Feds purchase makes credit available where no alternative existed. The composition effect will be captured by our second time period estimation (from 4/1/01 to 4/1/11) of Monetary Policy's effects as all of the credit easing policies employed by the Fed occurred over this time period. A little background on the implementation of these polices is introduced below.

Implementation of Credit Easing and Large Scale Asset Purchases*

*This section draws heavily from Sack 2010

The Federal Reserve holds the assets it purchases in the open market in its System Open Market Account (SOMA).  Historically, SOMA holdings have consisted of nearly all Treasury securities, although small amounts of agency debt have been held. Purchases and sales of SOMA assets are called outright open market operations (OMOs).  Outright OMOs, in conjunction with repurchase agreements and reverse repurchase agreements, traditionally were used to alter the supply of bank reserves in order to influence the federal funds rate.  Most of the higher-frequency adjustments to reserve supply were accomplished through repurchase and reverse repurchase agreements, with outright OMOs conducted periodically to accommodate trend growth in reserve demand. OMOs were designed to have a minimal effect on the prices of securities included in its operations.  It is the Fed's way of not distorting prices on debt instruments and thus protecting its independence from political pressure.  To this end, OMOs tended to be small in relation to the markets for Treasury bills and Treasury coupon securities. Large Scale Asset Purchases, however aimed to have a noticeable impact on the interest rates being purchased as well as on other assets with similar characteristics. In order to lower market interest rates, Large Scale Asset Purchases were designed to be large relative to the markets for these assets.  As mentioned in Gagnon, Raskin, Remache and Sack 2010:

Between December 2008 and March 2010, the Federal Reserve will have purchased more than $1.7 trillion in assets. This represents 22 percent of the $7.7 trillion stock of longer-term agency debt, fixed-rate agency MBS, and Treasury securities outstanding at the beginning of the LSAPs.

In the following discussion of the independent variables selected to capture this effect please note that they are all defined as Federal Reserves holdings as a percentage of the total market value outstanding.  In this way we can quantify how much the Fed's holdings relative to the total market supply of these assets impacted market risk premia.

Large Scale Asset Purchases were focused on four main securities:

1. Agency Debt

2. Mortgage Backed Securities

3. Treasury Securities

4. Commercial Paper

Although we do not explicitly account for these Treasury Purchases, we rely on our main balance sheet variable to capture their effects. The first asset to account for which is especially pertinent to our short-term risk premia variable is commercial paper.

Commercial Paper

Accounting for commercial paper and the Commercial Paper Funding Facility LLC, we will use the Fed's holdings as a percentage of the total commercial paper outstanding. The Commercial Paper Funding Facility LLC, like all of the Fed's Credit Easing tools was only functional during our second estimation period  (4/1/01 to 4/1/11). That is why it will only be used as a variable over that estimation period. Operationally:

Commercial Paper^{Fed}_{t}={CPaper^{Fed}_{t}\ CPaper_{t}^{total}}x 100

Where,

Commercial Paper^{Fed}_{t}= the percentage of the total commercial paper outstanding the Fed owns at time, t

CPaper^{Fed}_{t}= Net Portfolio Holdings of Commercial Paper Funding Facility LLC (WACPFFL) at time, t

CPaper_{t}^{total}= Commercial Paper Outstanding (COMPOUT) at time, t

We expect this variable to be negatively related to short-term risk premia over our estimation period. The reason being that increased Fed support in this market should have directly reduced the spread between commercial paper and Treasury bills.  Especially if the Fed sold T-bills to purchase short-term commercial paper and asset backed commercial paper.  Therefore the following hypothesis test is appropriate:

H_{0}:ß ≥ 0 vs. H_{a}: ß < 0 

With respect to the long-term risk premia, we should expect this monetary policy action to have a negligible effect.  This is because this policy was aimed at impacting short-term commercial paper and not longer-term rates:

H_{0}: ß = 0 vs. H_{a}: ß ≠ 0

Data Issues

The following data sets are pulled from FRED and their details are as follows:

(a) Assets - Net Portfolio Holdings of Commercial Paper Funding Facility LLC (DISCONTINUED SERIES) (WACPFFL), Weekly, As of Wednesday, Not Seasonally Adjusted, 2002-12-18 to 2010-08-25

(b) Commercial Paper Outstanding (COMPOUT), Weekly, Ending Wednesday, Seasonally Adjusted, 2001-01-03 to 2011-10-26

This required the following data transformation within FRED:

{(WACPFFL\1000)\ COMPOUT}x100

Mortgage-Backed Securities & Agency Debt

In order to account for the Feds holdings Agency Debt and Mortgage Backed Securities as a percentage of the total outstanding we use the following variable:

Agency Debt & MBS^{Fed}_{t} = {FADS^{Fed}_{t} + MBS^{Fed}_{t}\ DomesticFinancial_{t}^{Total}}x 100

Where, 

Agency Debt & MBS^{Fed}_{t}= Feds holdings of agency debt and Mortgage-Backed Securities as a percentage of the total outstanding at time, t

FADS^{Fed}_{t}= Fed's holdings of Federal Agency Debt Securities (WFEDSEC) at time, t

MBS^{Fed}_{t}= Fed's holdings of Mortgage-Backed Securities (WMBSEC) at time, t

DomesticFinancial_{t}^{Total}=  Domestic Financial Sectors holdings of Agency- and GSE-Backed Mortgage Pools (AGSEBMPTCMAHDFS) at time, t

This variable, theoretically should have almost no impact on both long-term and short-term risk premiums. The reason is Agency Debt and MBS are not highly correlated with either of our dependent variables, in fact it wasn't meant to impact these measures. It was however meant to influence 30 year mortgage rates which much research has shown it did in fact help ease.  We include this variable only because it was a major part of the Fed's credit easing policy and that future models with measures of housing affordability as their dependent variable would be able to use the variables listed in this paper to show Fed support of the housing market. 

The beta coefficient in front of this independent variable is therefore expected to have no significant relation to either long-term or short-term risk premiums as defined in this paper:

H_{0}: ß = 0 vs. H_{a}: ß ≠ 0

We fully expect to not reject the null hypothesis for both of our models.

Data Issues

The data for the above variables comes from the following financial time-series from FRED:

(a) Total Credit Market Assets Held by Domestic Financial Sectors - Agency- and GSE-Backed Mortgage Pools (AGSEBMPTCMAHDFS), Quarterly, End of Period, Not Seasonally Adjusted, 1949-10-01 to 2011-04-01

(b) Reserve Bank Credit - Securities Held Outright - Federal Agency Debt Securities (WFEDSEC), Weekly, Ending Wednesday, Not Seasonally Adjusted, 2002-12-18 to 2011-10-26

(c) Reserve Bank Credit - Securities Held Outright - Mortgage-Backed Securities (WMBSEC), Weekly, Ending Wednesday, Not Seasonally Adjusted, 2009-01-14 to 2011-10-26

Please keep dancing and wait for our next post which finishes defining our independent variables,

Steven J. 

Monetary Policy & Credit Easing pt. 2: Defining Our Variables

IN order to get a more complete picture of how monetary policy influences credit conditions we will estimate its effects on both long-term and short-term risk premia.  Our first dependent variable is the short-term risk premia and our second is the long-term risk premium. We will be testing the effects of monetary policy on both risk premias over two separate time periods. The first is from 4/1/71to 7/1/97 and the second is from 4/1/01 to 4/1/11.  We use two different time periods to more strongly capture the influence of the different monetary policy tools that were prevalent to each respective time period.  For example, the first time period was a time characterized by the Federal Reserve's indirect manipulation of the federal funds rate to influence other short-term rates like the prime bank loan rate and rates on short-term commercial paper.  In direct contrast, the second time periods estimation recognizes the Fed's manipulation of both the size and composition of its balance sheet as well as it's use of the federal funds rate to influence short-term market rates.

First Dependent Variable: Short-term Risk Premium & Commercial Paper

Commercial Paper is an unsecured promissory note with a fixed maturity of 1 to 270 days. We will be focusing on 90 day Commercial Paper. Commercial Paper is a money-market security issued by large banks and corporations to get money to meet short term debt obligations, and is only backed by an issuing bank or corporation's promise to pay the face amount on the maturity date specified on the note. Since it is not backed by collateral, only firms with excellent credit ratings from a recognized rating agency will be able to sell their Commercial Paper at a reasonable price. Additionally, Commercial Paper rates increase with maturity so they also have a duration risk associated with the price they fetch in the market place. Since this type of security is typically considered pretty risk free and has virtually zero rollover risk its deviation from the three-month Treasury bill rate seems like an appropriate measure of the short-term risk premium.  The 3 Month T-bill is used as our risk-free asset because it is considered to have zero default risk and is highly liquid. Moreover, T-bills are used for short-term financing purposes which makes its use very similar to that of Commercial Paper. The short-term risk premium is thus operationalized as follows:

SR^{premium}_{t}= CP3M_{t} – TB3MS_{t}

Where,

SR^{premium}_{t} = Short-term Risk Premium at time, t

CP3M_{t}= 3-Month Commercial Paper Rate at time, t

TB3MS_{t}=3-Month Treasury Bill: Secondary Market Rate at time, t

Data Issues

For the 3-Month Treasury Bill series we use the following from FRED:

(a) 3-Month Treasury Bill: Secondary Market Rate (TB3MS), Monthly, 1934-01-01 to 2011-09-01

The 3-Month Commercial paper series is unfortunately not so easy to deal with.  For one the series stops in 1997 and breaks off into two separate time series:

(b) 3-Month Commercial Paper Rate (DISCONTINUED SERIES) (CP3M), Monthly, 1971-04-01 to 1997-08-01

The two separate series include the financial commercial paper rate and the non-financial commercial paper rate:


(c) 3-Month AA Financial Commercial Paper Rate (CPF3M), Monthly, 1997-01-01 to 2011-09-01


(d) 3-Month AA Nonfinancial Commercial Paper Rate (CPN3M), Monthly, 1997-01-01 to 2011-09-01

To reconcile these issues, we take the average of the two and use them for the estimation of the Fed's policies over the second time period.

Second Dependent Variable: Long-term Risk Premium For Corporate Debt

For our long-term risk premium we choose to employ the 10 year Treasury Note rate as our risk-free rate because it shares the full promise of repayment by the United States Government.  Moody's Aaa rated securities aren't so lucky and therefore carry a risk premium associated with them. Although, the risk-premium for longer-term securities includes several things that are more acute under stress than our counterpart short-term risk premium. These include a heightened duration risk, liquidity risk and default risk.  We would expect our estimation of monetary policy effects on this variable to be more accurate as it theoretically should fluctuate more in response to actions taken by the Federal Reserve. The long-term risk premium is defined as follows:

LR^{premium}_{t}= BAA_{t} – GS10_{t}

where,

LR^{premium}_{t} =Long-term Risk Premium at time, t

BAA_{t} =Moody's Seasoned Baa Corporate Bond Yield at time, t

GS10_{t} =10-Year Treasury Constant Maturity Rate at time, t

Data Issues

All of the data here comes from FRED and there series details are listed as follows:

(a) Moody's Seasoned Baa Corporate Bond Yield (BAA), Monthly, 1919-01-01 to 2011-09-01


(b) 10-Year Treasury Constant Maturity Rate (GS10), Monthly, 1953-04-01 to 2011-09-01

Independent Variables: The Federal Reserve's Monetary Policy Toolbox

Our independent variables seek to capture the many tools the Federal Reserve can and has employed throughout its history.  This includes capturing the effects of traditionally unorthodox tools such as the manipulation of both the size (known as quantitative easing) and the composition (known as credit easing) of the Fed's balance sheet as well as capturing the effect from our more well known tools like changing the federal funds rate.  We will also seek to determine the effects of interest paid on reserves.


Accounting For The Size Of The Fed's Balance Sheet & Quantitative Easing

Our first and in the authors opinion most important independent variable seeks to capture the Fed's balance sheet effects on risk premiums.  It will be defined as the Feds holdings of credit market assets as a percentage of the total amount of assets held.  The more the Fed supports credit markets the larger this percentage will be.  It captures the balance sheets size as a percentage of the total market balance sheet. It is available over both our sample time periods and is therefore of pinnacle convenience to our analysis.  One special component of the balance sheet has been the holding of Treasury Securities. Before November of 2008, the Federal Reserve maintained a relatively small portfolio of between $700 billion and $800 billion in Treasury securities- an amount largely determined by the volume of dollar currency that was in circulation. In late November 2008 the Federal Reserve announced that it would purchase up to $600 billion of agency debt and agency mortgage-backed securities (MBS). In March 2009, it enlarged the program to include cumulative purchases of up to $1.75 trillion of agency debt, agency MBS, and longer-term Treasury securities. As mentioned previously, the use of the balance sheet for financial easing was initiated because the Federal Reserves main policy instrument, the federal funds rate had effectively reached the zero lower bound in late 2008.

Operationally we define this variables as:

FedBalance^{size}_{t}={CreditAssets^{Fed}_{t}\ CreditAssets_{t}^{total}}x 100

where,

FedBalance^{size}_{t}= the percentage of the total credit market assets the Fed owns at time, t

CreditAssets^{Fed}_{t}= Total Credit Market Assets Held by Domestic Financial Sectors - Monetary Authority (MATCMAHDFS) at time, t

CreditAssets_{t}^{total}= Total Credit Market Assets Held (TCMAH) at time, t

This variable is the percentage of the total credit market assets that the Fed holds. Its coefficient is meant to be negative so that as it increases market interest rate risk premiums decrease. It accounts for the effects of the size of the Fed's balance sheet. We expect this variable to have a negative effect on both short-term and long-term risk premia and therefore:

H_{0}: ß ≥ 0 vs. H_{a}: ß ≤ 0

Data issues

The data for this variable is available for extraction from FRED and are detailed as follows:

(a) Total Credit Market Assets Held (TCMAH), Quarterly, End of Period, Not Seasonally Adjusted, 1949-10-01 to 2011-04-01


(b) Total Credit Market Assets Held by Domestic Financial Sectors - Monetary Authority (MATCMAHDFS), Quarterly, End of Period, Not Seasonally Adjusted, 1949-10-01 to 2011-04-01

Monetary Policy & Credit Easing pt. 1: Background & Theoretical Considerations

An Introduction & Literary Review

Monetary Policy in the United States has traditionally been set to meet two objectives as defined in Federal Reserve Act; price stability and maximum employment.  In order to meet these goals the Federal Reserve manipulates the federal funds rate (FF) through a process called Open Market Operations (OMOs).  Unfortunately, when a recession is brought about by financial crisis this tool lose its potency and the economy enters into a "Liquidity Trap".  In a liquidity trap the FF is effectively at zero, and additional support is necessary to blunt the fall in asset prices and reduce measures of heightened financial stress. The Federal Reserve has recently enlisted a range of tools that are meant to provide further accommodation when their primary tool, the FF hits the lower bound.  These include manipulation of both the size and composition of its balance sheet, informational easing and paying interest on excess reserves.  We seek to formally investigate how these tools impact two important measures of financial stress, the long-term and short-term risk premia. 

There have been a slew of recent studies which seek to estimate the effects of Large Scale Asset Purchases (LSAP's) on Treasury Rates. Using an event-study methodology that exploits both daily and intra-day data, Krishnamurthy and Vissing-Jorgensen 2011 estimate the effects of both Quantitative Easing 1 and 2.  They find a large and significant drop in nominal interest rates on long-term safe assets (Treasuries, Agency bonds, and highly-rated corporate bonds).  

Sack, Gagnon, Raskin and Remache 2011 estimate the effects of large-scale asset purchases on the 10-year term preimium.  They use both an event-study methodology and a Dynamic OLS regression with Newey-West standard errors. They present evidence that the purchases led to economically meaningful and long-lasting reductions in longer-term interest rates on a range of securities, including securities that were not included in the purchase programs. Importantly, they find that these reductions in interest rates primarily reflect lower risk premiums, including term premiums, rather than lower expectations of future short-term interest rates.  

In 1966 Franco Modigliani and Richard Sutch wrote a seminal piece on Monetary Policy titled ``Innovations in Interest Rate Policy." In the paper the authors estimate the effects of ``Operation Twist", a policy by the Federal Reserve and the Kennedy Administration aimed at affecting the term structure of the yield curve.  In summary they find that the targeting of longer maturities has a rather minimum effect on the spread between short-term and long-term government debt securities.    

Bernanke, Reinhart and Sack 2004, estimate the effects of ``non-standard policies" when the Federal Funds Rate hits the lower bound.  They find that the communications policy can be used to effectively lower long-term yields when short-term interest rates are trapped at zero. They also find evidence supporting the view that asset purchases in large volume by a central bank would be able to affect the price or yield of the targeted asset.  This research was most likely the basis for the Feds actions taken over the course of the latest U.S. financial crises.  

Theoretical Model, Assumptions & Further Details

A risk premium is the amount a debt issuer has to pay in order to borrow above the interest rates on the safest of assets for a given maturity, m. By comparing interest rates on debt with the same maturity we are able to isolate the part of the spread that stems from duration risk from the other factors that influence the risk of default.  Additionally, by using only nominal debt instruments we remove elements in the spread that stem from inflation compensation.  

Risk premiums are thus defined as follows:

r_{m}^{premium} = r^{RR}_{m} - r^{Rf}_{m} 

Where,

 r_{m}^{premium} = Risk Premium for time till maturity, m

r^{RR}_{m}= Risky interest rate on nominal debt, for time till maturity, m

 r^{Rf}_{m} = Risk free interest rate for nominal debt, for time till maturity, m

In order to see what factors influence r_{m}^{premium} we have to analyze what moves the interest paid on the risk free interest rate, r^{Rf}_{m}, which is usually defined as some sort of United States Government debt, and the interest rate that carries risk, r^{RR}_{m}.  

Uncertainty and financial stress go hand in hand as  well documented in Charles P. Kindleberger's "Manias, Panics and Financial Crisis".  Historically during periods of high uncertainty, asset prices fluctuate wildly as more cautious investors cling to the safest assets (known as flight to safety) and the more bold investors bargain shop. Investors sell assets that carry r^{RR}_{m} and purchase those that carry r^{Rf}_{m}.  This causes the r_{m}^{premium} to increase dramatically and it becomes relatively more expensive for firms to access the capital markets to meet their funding needs.  There is a shortage of credit or credit crunch as debt issuers struggle to find buyers of their debt. 

In expansionary times the two interest rates that determine the risk premium move towards each other thus decreasing the risk premia.  Investors feel more confident and become hungry for yield, this leads to movement away from the risk-less lower interest carrying assets into riskier assets with a higher yield.  This pushes down the yield on the riskier assets and pushes the yield on the riskless assets up, thus making the return on these assets similar.   

Room For Policy

During periods of financial stress the Federal Reserve can reduce the risk premia and thus ease credit conditions by moving either r^{Rf}_{m} or r^{RR}_{m}.  The Fed has relied on the "portfolio balance channel" in order to reduce the financial stress felt by credit worthy firms.  As the Fed purchases Treasuries, yield hungry and  "crowded out" investors may purchase assets with similar credit ratings (like bonds with a AAA rating) in order to capture that increased yield differential thus lowing the yield on these assets.  

Brian P. Sack, Executive Vice President of the Federal Reserve, provided a great description of the Portfolio Balance Channel in a 2010 speech given at the CFA Institute Fixed Income Management Conference:

Under that view (portfolio balance channel view), our (the Fed) asset holdings keep longer-term interest rates lower than otherwise by reducing the aggregate amount of risk that the private markets have to bear. In particular, by purchasing longer-term securities, the Federal Reserve removes duration risk from the market, which should help to reduce the term premium that investors demand for holding longer-term securities. That effect should in turn boost other asset prices, as those investors displaced by the Fed’s purchases would likely seek to hold alternative types of securities.

All other things being equal the risk premia should decrease because the U.S. Treasury market is the most liquid market on earth. So the decrease in Treasury yields should be less than that of the less-liquid and risk-bearing assets.  

The Fed can also influence the risk premia by purchasing the risk bearing asset directly.  Examples of this include its implementation of the Commercial Paper Funding Facility (CCFF) and Agency Mortgage-Backed Securities Purchase Program (ABMBSPP).

Credit Easing is another channel the Federal Reserve has looked to exploit. Credit Easing policies involve changing the composition of the Fed's Balance Sheet from risk-less assets to riskier ones, all while keeping its size constant. Operationally it involves selling risk-free assets like 3 month T-bills to finance the purchase of risk-bearing assets like 3 month Commercial Paper. These assets have the same maturity m and the goal of the operation is accomplished as it circumvents the need to reduce the size of balance sheet. These policies lead to lower risk premiums as they increase the rate r^{Rf}_{m} on the risk-free asset being sold and decrease the r^{RR}_{m}, or interest rate on the risky asset being bought in the risk-free assets place.  This leads to additional easing as investors feel more certain that the market value of these assets will be supported by the Fed's holdings.  Removing the uncertainty leads to these riskier assets being transformed into less risky ones, thus increasing the appeal for them in periods of tumultuous financial stress.

In the next post we will delve into defining our dependent variables which seek to explicitly capture in risk premia while also looking at a few of our independent variables.  

Please people keep dancing into the new year,

Steven J.

Monetary Policy and Credit Easing

Here at the dancing economist, we wish to educate our followers on the finer points of economics and this includes econometrics and using R. R as mentioned previously is a free statistical software that enables regular people like us to do high end economics research. Recently, I wrote a paper on how the Federal Reserves actions have impacted both short-term and long-term risk premiums. In the next few blog posts I will be posting sections of the paper along with the R code necessary to perform the statistical analysis involved. One interesting result is that the Feds balance sheet although not previously manipulated was heavily involved in reducing long-term risk premia over the period from 1971 to 1997. The methodology in the paper involved performing a Generalized Least Squares procedure and accounting for residual correlation to achieve the assumptions as stated by the Gauss-Markov Theorem. More will follow,


Keep Dancing,

Steven J.

Janet Yellen and the Fed's New Approach to Asset Bubbly

     I love Janet Yellen because she is an academic and a Hyman P. Minsky follower on the Federal Reserves Board of Governors.  In a recent speech titled "The Federal Reserve's Asset Purchase Program", Yellen breaks down the Fed's decision to engage in a second round of asset purchases while offering the best research available.  She delves into the beverage curve- which is the relationship between unemployment and job vacancies- to talk about both structural and cyclical factors affecting unemployment.  Additionally, she details the Fed's new attempts at monitoring quantitative easing effects on credit flows and asset bubbles.  This is a revolutionary and remarkable change in procedure at the Fed because before this crises the Fed would have completely denied looking at asset price movements and credit in making its decisions.  This signals a significant shift in the attitude and mindset that recognizes that markets are not always efficient or rational, especially when it comes to asset prices.  In evaluating the dangers of the Fed's latest round of quantitative easing, Yellen mentioned the following asset price and credit indicators:

With respect to the stock market:

"In the stock market, for example, price-to-earnings ratios, by some measures, remain below their averages over the past several decades, and other valuation measures also indicate that equity prices are not significantly out of alignment with past norms."

Then looking at the Price-to-rent ratio for the housing market:

"In the real estate market, price-to-rent ratios for both residential and commercial real estate are now within a reasonable range of their long-run averages, in contrast to the severe misalignment that occurred prior to the crisis. Again, there is little sign here of imbalances relative to fundamentals, at least if history is used as a guide."

In the Bond market we have seen the obvious bubble created by the fed when it intervenes into the market and buys up treasuries:

"In fixed-income markets, narrow risk spreads and risk premiums could be signs of excessive risk-taking by investors, and indeed spreads on corporate bonds have dropped dramatically since the financial crisis, as the economic outlook has improved and investor sentiment has picked up. Risk premiums on nonfinancial corporate bonds, as measured by forward spreads far in the future, are relatively low compared with historical norms, although other indicators for this market do not point to overvaluation." 

Yellen even mentions identifying financial imbalances by focusing more directly on measuring credit flows and exposure to credit risk! In this area the risks are mute:

"Thus, there is little evidence that financial institutions are significantly expanding the level of credit and liquidity provided to households and businesses on net. Indeed, given the current very low level of interest rates and the continuation of the economic recovery, credit flows remain stubbornly sluggish."

The Fed even created a new survey called the Senior Credit Officer Opinion Survey on Dealer Financing Terms to monitor leverage:

"To monitor leverage provided by dealers to financial market participants, last June the Federal Reserve launched the Senior Credit Officer Opinion Survey on Dealer Financing Terms. This survey provides information on credit terms and availability of various forms of dealer-intermediated financing, including funding for securities positions and over-the-counter derivatives. The survey results suggest that over the past several months there has been some easing of terms applicable to financing for a range of counterparty types and many types of collateral, as well as an increase in demand from clients to fund most types of securities. These results indicate that the availability and use of leverage by nonbank financial institutions increased somewhat last year."

The Fed is on track to be extremely successful in the long-run.  Although, the additional asset purchases are a dangerous move given that the Fed has created and set in motion a bond bubble.  The asset purchases explicitly say to the financial markets that the Fed will not be afraid to use this tool in the future, which may in itself create a moral hazard for the bond market.  The thing that I love best about Janet Yellen's speech must be the explicit statement about using adaptive regulation to target financial imbalances versus using the extremely blunt federal funds rate.  Janet wants to work closely with other regulators to monitor systemic risk and nip asset inflation in the bud.  I wrote about this a while back (June 21,2010 to be exact)  thus making me especially delighted that someone on the board has formed the same views (albeit independently):

"We are working with other regulators to make the financial system more robust and are attentive in our supervision to developments that may affect systemic risk. If evidence of financial imbalances were to develop, I believe that supervision and regulation should provide the first line of defense so that monetary policy can concentrate on its longstanding goals of price stability and maximum employment. That said, we cannot categorically rule out using monetary policy to address financial imbalances, given the damage that they can cause."

A graphical illustration of quantitative easing & the bond market rejection

On November 3, 2010 the F.O.M.C. announced that it would be purchasing 600 billion in treasuries on a schedule of $75 billion a month till the second quarter of 2011.  When the Fed does quantitative easing it is mostly just purchasing 2 to 10 year nominal U.S. treasury bonds.  The main channel through which quantitative easing is supposed to work is called the portfolio effect.  This works as follows, the Fed purchases treasuries therefore reducing the total supply on the open market.  The supply curve for treasuries in figure 1 shifts to the left from BS0 to BS1 as we move from A to B. Notice that their yield or the interest rate return falls from R0 to R1.  The fall in this interest rate causes investors to seek higher returns in assets with comparable maturities and interest rate risk structure.  A 10 year AAA corporate bonds yield would then be more attractive relative to 10 yr nominal treasuries and their demand curve in figure 2 would shift right from DCB0 to DCB1.  The lower interest rate on these corporate bonds would make it easier for firms with access to the capital markets to issue debt for long term business investment.

It is now January 4^th, 2011 and we wish to see if the fed’s plan to spur long term investment is doing what the portfolio effect would suggest.  Figure A is the same as before with the fed reducing the total available supply of treasuries on the market. 

 

Figure B & C: The bond market rejects the purchases so the demand for treasuries shifts to the left from BD0 to BD1 and we move from point B to point C in figure B.  The Feds actions are completely negated in the treasury market.  Since AAA rated corporate bonds are closely tied to treasuries, the portfolio effect has also worked the other way.  This was witnessed when the demand for corporate bonds shifted to the left.  This is represented in figure C with the movement from A to B and a higher interest rate as the demand curve shifted from DCB0 to DCB1.  The higher interest rate discourages debt issuance for the support of long term investment.

As one can see from the following chart, this is exactly what occurred.  The bond market rejected the fed’s actions and started a massive sell off which lead to the steepening of the yield curve.

The following figure D, wraps up the story.  With the Fed agreeing to purchase 600 billion more treasuries, people may be feeling that bonds are overpriced and decided to take their money elsewhere, mainly the stock market.  The less risk adverse attitude and hope for a higher return led people to dump their bonds and buy other assets like equities.  The demand for equities in the diagram below shifts to the right from D0 to D1 as it moves from point A to B. 

The following chart describes that this is exactly what happened as the yield curve steepened so did the S&P rage.