AR1.RPF

AR1.RPF is an example of estimation of a model with AR(1) errors, using the AR1 instruction with several different options for handling the serial correlation. It also includes the alternative of estimation of LINREG with HAC standard errors.

(This is an example from textbook, so the data series are just labeled X and Y).

The simple regression

linreg y
# constant x

gives a very low Durbin-Watson, indicating rather strongly serially correlated errors. With such a small D-W, a regression in first differences is one possibility. That is most easily done using AR1 with an input value of RHO=1.0.

ar1(rho=1.0) y
# constant x

Because first differencing zeros out the CONSTANT, the output shows it as a zero coefficient with zero standard error.

There are several options for estimating the \(\rho\). The next two instructions do the "conditional" estimates, which drop the first observation: HILU (grid search) and CORC (iterated).

ar1(method=hilu) y
# constant x
ar1(method=corc) y
# constant x

As one would hope, HILU and CORC give (effectively) identical results. The next instruction does maximum likelihood (MAXL option)

ar1(method=maxl) y
# constant x

which comes in with a somewhat higher value of .95 vs .89. This isn’t an unreasonable difference given that there are only 40 data points, and MAXL can use all of them while CORC can only use 39.

The final set of estimates redoes the least squares regression, but corrects the covariance matrix for HAC standard errors

linreg(robust,lwindow=neweywest,lags=4) y
# constant x

The point estimates are the same as the original LINREG, but the standard errors are quite a bit higher.

Full Program

cal(a) 1959
open data ar1.prn
data(format=prn,org=columns) 1959:1 1998:1
*
* OLS regression
*
linreg y
# constant x
*
* First difference regression. This may be the best choice if the
* autocorrelation coefficient is close to 1. (Note that the
* coefficient on the CONSTANT gets zeroed out, since it's
* eliminated by first differencing).
*
ar1(rho=1.0) y
# constant x
*
* AR1 regression using several methods. HILU and CORC should give
* almost identical answers unless there are multiple roots.
*
ar1(method=hilu) y
# constant x
ar1(method=corc) y
# constant x
ar1(method=maxl) y
# constant x
*
* OLS with Newey-West standard errors. This allows for
* autocorrelation of up to four lags.
*
linreg(robust,lwindow=neweywest,lags=4) y
# constant x

Output

Linear Regression - Estimation by Least Squares

Dependent Variable Y

Annual Data From 1959:01 To 1998:01

Usable Observations 40

Degrees of Freedom 38

Centered R^2 0.9584495

R-Bar^2 0.9573561

Uncentered R^2 0.9990931

Mean of Dependent Variable 85.645000000

Std Error of Dependent Variable 12.956316151

Standard Error of Estimate 2.675532533

Sum of Squared Residuals 272.02202467

Regression F(1,38) 876.5495

Significance Level of F 0.0000000

Log Likelihood -95.0976

Durbin-Watson Statistic 0.1229

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant 29.519254786 1.942346872 15.19773 0.00000000

2. X 0.713659422 0.024104758 29.60658 0.00000000

Regression with AR1 - Estimation by Input Value of Rho

Dependent Variable Y

Annual Data From 1960:01 To 1998:01

Usable Observations 39

Degrees of Freedom 37

Centered R^2 0.9942278

R-Bar^2 0.9940718

Uncentered R^2 0.9998873

Mean of Dependent Variable 86.341025641

Std Error of Dependent Variable 12.344862941

Standard Error of Estimate 0.950490737

Sum of Squared Residuals 33.427007715

Log Likelihood -52.3318

Durbin-Watson Statistic 1.5097

Q(9-1) 7.5923

Significance Level of Q 0.4742690

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant 0.0000000000 0.0000000000 0.00000 0.00000000

2. X 0.7199555154 0.0792433390 9.08538 0.00000000

3. RHO 1.0000000000 0.0000000000 0.00000 0.00000000

Regression with AR1 - Estimation by Hildreth-Lu Search

Dependent Variable Y

Annual Data From 1960:01 To 1998:01

Usable Observations 39

Degrees of Freedom 36

Centered R^2 0.9953392

R-Bar^2 0.9950802

Uncentered R^2 0.9999090

Mean of Dependent Variable 86.341025641

Std Error of Dependent Variable 12.344862941

Standard Error of Estimate 0.865881117

Sum of Squared Residuals 26.991003917

Regression F(2,36) 3843.9763

Significance Level of F 0.0000000

Log Likelihood -48.1615

Durbin-Watson Statistic 1.6040

Q(9-1) 9.2616

Significance Level of Q 0.3207097

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant 45.039294676 6.243781141 7.21346 0.00000002

2. X 0.550888733 0.065886010 8.36124 0.00000000

3. RHO 0.891345256 0.042582756 20.93207 0.00000000

Regression with AR1 - Estimation by Cochrane-Orcutt

Dependent Variable Y

Annual Data From 1960:01 To 1998:01

Usable Observations 39

Degrees of Freedom 36

Centered R^2 0.9953392

R-Bar^2 0.9950802

Uncentered R^2 0.9999090

Mean of Dependent Variable 86.341025641

Std Error of Dependent Variable 12.344862941

Standard Error of Estimate 0.865881117

Sum of Squared Residuals 26.991003939

Regression F(2,36) 3843.9763

Significance Level of F 0.0000000

Log Likelihood -48.1615

Durbin-Watson Statistic 1.6040

Q(9-1) 9.2620

Significance Level of Q 0.3206810

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant 45.040044636 6.244136857 7.21317 0.00000002

2. X 0.550882475 0.065889102 8.36075 0.00000000

3. RHO 0.891351908 0.042582035 20.93258 0.00000000

Regression with AR1 - Estimation by Beach-MacKinnon

Dependent Variable Y

Annual Data From 1959:01 To 1998:01

Usable Observations 40

Degrees of Freedom 37

Centered R^2 0.9949256

R-Bar^2 0.9946513

Uncentered R^2 0.9998892

Mean of Dependent Variable 85.645000000

Std Error of Dependent Variable 12.956316151

Standard Error of Estimate 0.947557352

Sum of Squared Residuals 33.221002591

Log Likelihood -54.2141

Durbin-Watson Statistic 1.5175

Q(10-1) 7.6982

Significance Level of Q 0.5648281

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant 26.673429237 5.103987853 5.22600 0.00000700

2. X 0.724927554 0.058683004 12.35328 0.00000000

3. RHO 0.950664003 0.049938333 19.03676 0.00000000

Linear Regression - Estimation by Least Squares

HAC Standard Errors with Newey-West/Bartlett Window and 4 Lags

Dependent Variable Y

Annual Data From 1959:01 To 1998:01

Usable Observations 40

Degrees of Freedom 38

Centered R^2 0.9584495

R-Bar^2 0.9573561

Uncentered R^2 0.9990931

Mean of Dependent Variable 85.645000000

Std Error of Dependent Variable 12.956316151

Standard Error of Estimate 2.675532533

Sum of Squared Residuals 272.02202467

Log Likelihood -95.0976

Durbin-Watson Statistic 0.1229

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant 29.519254786 4.298313417 6.86764 0.00000000

2. X 0.713659422 0.053380966 13.36917 0.00000000