## TA: Nicolas Bottan

Welcome to the seventh tutorial of Econ 508. In the current issue I am going to summarize some well known tests for autocorrelation and ARCH processes. I draw on Johnston and DiNardo’s (1997) Econometric Methods, and Professor Koenker’s Lecture 7. We also provide additional support on testing for heteroscedasticity (see Appendix) and a starting point for those who want to explore further aspects of ARCH and GARCH process (see Perrelli, 2001). 1

# Tests for Autocorrelated Errors

### Background:

If you run a regression without lagged variables, and detect autocorrelation, your OLS estimators are unbiased, consistent, but inefficient and provide incorrect standard errors. In the case that you include lagged dependent variables among the covariates and still detect autocorrelation, then you are in bigger trouble: OLS estimators are inconsistent.

To test for the presence of autocorrelation, you have a large menu of options. Here we suggest the use of the Breusch-Godfrey test, and we will show how to implement this test using the dataset AUTO2.dta, which you can download from the Econ 508 web site (Data). As you will see, this adapted data set contains five series.

This time we will load the dataset directly from the course website:

    webuse set "http://www.econ.uiuc.edu/~econ508/data/"    webuse "AUTO2.DTA", clear    list in 1/6
     +-------------------------------------------------------+     | quarter         gas      price      income      miles |     |-------------------------------------------------------|  1. |  1959.1   -8.015248    4.67575    -4.50524   2.647592 |  2. |  1959.2    -8.01106   4.691292   -4.492739   2.647592 |  3. |  1959.3   -8.019878   4.689134   -4.498873   2.647592 |  4. |  1959.4   -8.012581   4.722338   -4.491904   2.647592 |  5. |  1960.1   -8.016769    4.70747   -4.490103   2.647415 |     |-------------------------------------------------------|  6. |  1960.2   -7.976376   4.699136   -4.489107   2.647238 |     +-------------------------------------------------------+ 

As we did before we need to transform the data in “time series” first:

    sort quarter     gen t=_n      label variable t "Integer time period"      tsset t  
time variable:  t, 1 to 128       delta:  1 unit
    gen price2= price*price      gen priceinc= price*income   

## Test: Breusch-Godfrey

### Background:

Suppose you are running a version of model (2), problem set 2, in which the original data is replaced by AUTO2. Then your model will be:

$gas_{t} = \beta_{0} + \beta_{1} income_{t}+ \beta_{2} price_{t} + \beta_{3} (price_{t})^{2} + \beta_{4} (price_{t}*income_{t}) + u_{t}$

and you wish to test whether the disturbances are autocorrelated. The steps to do that are as follows:

1. Run an OLS in your original equation:
    qui: regress gas income price price2 priceinc 

1. Obtain the estimated residuals:
    predict uhat, resid
1. Regress the estimated residuals (uhat) on the explanatory variables of the original model (income, price, price2, priceinc, constant) and lagged residuals (L.uhat). Call this the auxiliary regression.
    qui: regress uhat income price price2 priceinc L.uhat 
1. From the auxiliary regression above, obtain the R-squared and multiply it by the number of included observations:
    scalar nR2 = e(N)*e(r2)    display nR2
115.45707
1. Under the null hypothesis of no autocorrelation, the test statistic NR2 converges asymptotically to a Chi-squared with s degrees of freedom, where s is the number of lags of the residuals included in the auxiliary regression. In the case above, s=1, and we have:
    display "The Chi^2 critical value is: " invchi2(1,.95)

The Chi^2 critical value is: 3.8414588

In the example above, NR2 = 115.46 > 3.84 = Chi2 (1, 5%). Hence, we reject the null hypothesis of no autocorrelation on the disturbances.

# Test for ARCH Errors

To test for ARCH errors, you can use an LM test as follows:

1. Run an OLS in your original equation:
    qui: regress gas income price price2 priceinc
1. Generate the residuals and the squared residuals.
    predict vhat, resid    gen vhat2 = vhat^2
1. Regress squared residuals on the explanatory variables of the original model (income, price, price2, priceinc, constant) and lagged squared residuals. Call this an auxiliary regression.
    qui: regress vhat2 L.vhat2 L2.vhat2 L3.vhat2 L4.vhat2 income price price2 priceinc
1. From the auxiliary regression, calculate NR2 and compare with a Chi-squared (q, 5%), where q is the number of included lags of the squared residuals:
    scalar nR2 = e(N)*e(r2)    display "n*R2= " nR2 " and the Chi2 critical value is: " invchi2(4,.95)
n*R2= 89.061572 and the Chi2 critical value is: 9.487729

Under the null hypothesis of no ARCH errors, the test statistic NR2 converges asymptotically to a Chi-squared with q degrees of freedom, where q is the number of lags of the squared residuals included in the auxiliary regression. In the case above, q=4, and NR2=89.06 > 9.49 = Chi-squared(4, 5%). Therefore, we reject the null hypothesis of no ARCH, and admit that our regression presents time-varying variance.

# Appendix: Tests for Heteroscedasticity

Under heteroscedastic errors, it is well known that OLS estimators are unbiased and consistent, but inefficient and provide incorrect standard errors. Hence it is very important to detect this anomaly in your regression.

We will illustrate how to test for heteroscedasticity using Current Population Survey (CPS) data consisting on 100 observations on wages, educational level, years of experience, and unionization status of U.S. male workers. The data was borrowed from J&DN’s (1997) Econometric Methods, and slightly adjusted for the purposes of this tutorial. The variables are defined as follows:

Variable Description
lnwage log of hourly wage in dollars
exp Years of experience
union Dummy variable: 1 if union member, 0 otherwise

    webuse "CPS.dta", clear    list in 1/6
     +--------------------------------+     |   lnwage   grade   exp   union |     |--------------------------------|  1. | 2.331172       8    22       0 |  2. | 1.504077      14     2       0 |  3. | 3.911523      16    22       0 |  4. | 2.197225       8    34       1 |  5. | 2.788093       9    47       0 |     |--------------------------------|  6. | 2.351375       9    32       0 |     +--------------------------------+
    gen exp2=exp^2

After you download the data, the next step is to run a “traditional” wages equation involving the variables above described. In R, you can do that as follows:

    regress lnwage grade exp exp2 union

Source |       SS       df       MS              Number of obs =     100-------------+------------------------------           F(  4,    95) =    0.56       Model |  2.3434e+69     4  5.8585e+68           Prob > F      =  0.6957    Residual |  1.0024e+71    95  1.0552e+69           R-squared     =  0.0228-------------+------------------------------           Adj R-squared = -0.0183       Total |  1.0259e+71    99  1.0362e+69           Root MSE      =  3.2e+34------------------------------------------------------------------------------      lnwage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]-------------+----------------------------------------------------------------       grade |   8.91e+31   9.56e+31     0.93   0.354    -1.01e+32    2.79e+32         exp |   9.55e+31   7.49e+31     1.28   0.205    -5.31e+31    2.44e+32        exp2 |   7.88e+29   9.27e+29     0.85   0.398    -1.05e+30    2.63e+30       union |  -3.55e+31   1.07e+32    -0.33   0.741    -2.48e+32    1.77e+32       _cons |  -2.68e+33   5.92e+33    -0.45   0.652    -1.44e+34    9.08e+33------------------------------------------------------------------------------

## Test 1: White

Here the strategy is as follows:

1. Run the OLS regression (as you’ve done above, the results are omitted):
    qui: regress lnwage grade exp exp2 union
1. Get the residuals:
    predict resid, resid
1. Generate the squared residuals:
    gen resid2 = resid^2
1. Generate new explanatory variables, in the form of the squares of the explanatory variables and the cross-product of the explanatory variables:
    gen grade2 = grade^2     gen exp4 = exp2^2     gen gradexp = grade*exp     gen gradexp2 = grade*exp2     gen gradeuni = grade*union     gen exp3 = exp*exp2     gen expunion = exp*union     gen exp2uni = exp2*union

Because union is a dummy variable, its squared values are equal to the original values, and we don’t need to add the squared dummy in the model. Also the squared experience was already in the original model (in the form of exp2), so we don’t need to add that in this auxiliary regression.

1. Regress the squared residuals into a constant, the original explanatory variables, and the set of auxiliary explanatory variables (squares and cross-products) you’ve just created:
    qui: regress resid2 grade exp exp2 union grade2 exp4 exp3 gradexp gradexp2 gradeuni     expunion exp2uni
1. Get the sample size (N) and the R-squared (R2), and construct the test statistic N*R2:
    scalar nR2 = e(N)*e(r2)    display nR2
10.788134
1. Under the null hypothesis, the errors are homoscedastic, and NR2 is asymptotically distributed as a Chi-squared with k-1 degrees of freedom (where k is the number of coefficients on the auxiliary regression). In this last case, k=13.

And we observe that the test statistic NR2 is about 10.79, while the Chi-squared(12, 5%) is about 21.03, much bigger than the test statistic. Hence, the null hypothesis (homoscedasticity) can not be rejected.

## Test 2: Breusch-Pagan-Godfrey

The Lagrange Multiplier test proposed by BPG can be executed as follows:

1. Run the OLS regression (as you’ve done above, the output is omitted):
    qui: regress lnwage grade exp exp2 union
1. Get the sum of the squared residuals:
    predict error, resid     matrix accum E=error    matrix list E* Or obtain directly from regression output    dis e(rss)
symmetric E[2,2]           error      _conserror  20.989384_cons  4.470e-08        100
1. Generate a disturbance correction factor in the form of sum of the squared residuals divided by the sample size:
    scalar sigmahat=e(rss)/e(N)    dis sigmahat
.20989384
1. Regress the adjusted squared errors (in the form of original squared errors divided by the correction factor) on a list of explanatory variables supposed to influence the heteroscedasticity. Following JDN, we will assume that, from the original dataset, only the main variables grade, exp, and union affect the heteroscedasticity. Hence:
    gen adjerr2=(error^2)/sigmahat     regress adjerr2 grade exp union
      Source |       SS       df       MS              Number of obs =     100-------------+------------------------------           F(  3,    96) =    1.43       Model |  10.7047726     3  3.56825754           Prob > F      =  0.2386    Residual |  239.425216    96  2.49401266           R-squared     =  0.0428-------------+------------------------------           Adj R-squared =  0.0129       Total |  250.129988    99  2.52656554           Root MSE      =  1.5792------------------------------------------------------------------------------     adjerr2 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]-------------+----------------------------------------------------------------       grade |   .0989441   .0643511     1.54   0.127    -.0287919    .2266801         exp |   .0099537   .0131975     0.75   0.453    -.0162432    .0361506       union |  -.5824294   .3963325    -1.47   0.145    -1.369143    .2042844       _cons |  -.3260997   .9492019    -0.34   0.732    -2.210251    1.558051------------------------------------------------------------------------------

This auxiliary regression gives you a model sum of squares (ESS):

    scalar ESS=e(mss) 
1. Under the null hypothesis of homoscedasticity, (1/2) ESS asymptotically converges to a Chi-squared(k-1, 5%), where k is the number of coefficients on the auxiliary regression. In the last case, k=4. Hence, we need to compare (1/2) ESS with a Chi-squared with 3 degrees of freedom and 5%. Doing so we get (1/2) ESS = 5.35, while the critical value of a Chi-squared (3, 5%) = 7.81. Therefore, the test statistic falls short of the critical value, and the null hypothesis of homoscedasticity can not be rejected.

## Test 3: Goldfeld-Quandt

Suppose now you believe a single explanatory variable is responsible for most of the heteroscedasticy in your model. For example, let’s say that experience (exp) is the “trouble-maker” variable. Hence, you can proceed with the Goldfeld-Quandt test as follows:

1. Sort your data according to the variable exp. Then divide your data in, say, three parts, drop the observations of the central part, and run separate regressions for the bottom part (Regression 1) and the top part (Regression 2). After each regression, ask for the respective Residual Sum of Squares RSS:

2. Then compute the ratio of the Residuals Sum of Squares, R= RSS2/RSS1. Under the null hypothesis of homoscedasticity, this ratio R is distributed according to a $$F_{\left(\frac{(n-c-2k)}{2},\frac{(n-c-2k)}{2}\right)}$$, where n is the sample size, c is the number of dropped observations, and k is the number of regressors in the model.

This is left for the reader as an exercise. To check your results you should get: $$R < F$$, and as a consequence can not reject the null hypothesis of homocedasticity