**The Dancing Economist**, and kindly contributed to R-bloggers)

**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**

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

*>yc1<-earl[,”yc”]*

**1. Estimate the OLS regression**

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

*> 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*

**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.

*> library(nlme)*

*>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)*

*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*

**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*

*> library(FitAR)*

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

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

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

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

**leave a comment**for the author, please follow the link and comment on his blog:

**The Dancing Economist**.

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