SHAZAM The Chow Test

## Testing for Structural Stability - the Chow Test

It may be of interest to test for stability of regression coefficients between two periods. A change in parameters between two periods is an indication of structural change. Following an OLS estimation, the Chow test statistic for structural change is reported with the commands:

 ```OLS . . . DIAGNOS / CHOWONE=n1 ```

where `n1` is the number of observations in the first group.

Alternatively, to get test statistics computed for every breakpoint in the data set the following commands can be used.

 ```OLS . . . DIAGNOS / CHOWTEST ```

The computations required to obtain the Chow test statistic can be illustrated with SHAZAM commands. These commands give an example of programming in SHAZAM.

#### Example

This example is from Exercise 8.35 of Gujarati [1995, pp. 279-280]. A data set on personal savings and income for the United States is available for the years 1970 to 1991. It is of interest to investigate if there is a significant change in the savings-income relationship for the period 1970-1980 and 1981-1991 (the Reagan-Bush presidency era).

Gujarati suggests that either a linear or log-linear model may be used to estimate a savings-income relationship. The SHAZAM commands (filename: `USECON.SHA`) below estimate both a linear and log-linear model. After each OLS estimation a Chow test for structural change is computed.

```SAMPLE 1 22
* Estimate the savings-income relationship
OLS SAVINGS INCOME / RSTAT DWPVALUE
DIAGNOS / CHOWONE=11

* Now consider a log-linear relationship
GENR LSAV=LOG(SAVINGS)
GENR LINC=LOG(INCOME)
OLS LSAV LINC / LOGLOG RSTAT DWPVALUE
DIAGNOS / CHOWONE=11
STOP
```

The SHAZAM output can be viewed. For the linear savings-income function the Chow test statistic is reported as:

```SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS
N1   N2    SSE1        SSE2       CHOW    PVALUE    G-Q       DF1  DF2 PVALUE
11   11  1010.8      5103.5      20.371    0.000  0.1981        9    9 0.012

CHOW TEST - F DISTRIBUTION WITH DF1=   2 AND DF2=  18
```

For the log-linear savings-income function the Chow test statistic is reported as:

```SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS
N1   N2    SSE1        SSE2       CHOW    PVALUE    G-Q       DF1  DF2 PVALUE
11   11 0.11833     0.15911      13.923    0.000  0.7437        9    9 0.333

CHOW TEST - F DISTRIBUTION WITH DF1=   2 AND DF2=  18
```

The above SHAZAM output uses the following notation:

 `N1` no. of observations in group 1 `N2` no. of observations in group 2 `SSE1` sum of squared errors from a regression for group 1 `SSE2` sum of squared errors from a regression for group 2 `CHOW` the Chow test statistic `G-Q` the Goldfeld-Quandt test statistic for testing for equality of error variance in the 2 groups.   `G-Q = (SSE1/DF1)/(SSE2/DF2)`

By inspecting the output it can be seen that for both the linear and log-linear model the p-value reported for the Chow test statistic is less than 0.0005. This gives evidence to reject the null hypothesis of equality of regression coefficients in the 2 periods.

It should be considered that any test statistic relies on some distributional assumptions. The derivation of the Chow test assumes that the errors have the same variance (homoskedasticity) in the 2 groups and the errors are independently distributed (that is, no autocorrelation). Are these assumptions reasonable for this example ?

The SHAZAM output for the Chow test statistic also reports the Goldfeld Quandt test statistic for equal variance in the 2 groups. The above output shows that for both the linear and the log-linear model the calculated test statistic is less than 1. The p-value that is reported at the extreme right of the SHAZAM output is the p-value for a test of the null hypothesis of equal variance against the alternative hypothesis of larger variance in the second group compared to the first group. The results show that there is evidence for heteroskedasticity in the linear model. However, the homoskedasticity assumption appears reasonable for the log-linear model.

For both models the Durbin-Watson test statistic rejects the null hypothesis of no autocorrelation in the errors. Therefore, there is evidence for model misspecification.

It may be reasonable to consider that savings behaviour is related to savings in the past. This can be recognized by including a lagged dependent variable as an explanatory variable in the regression equation. The next list of SHAZAM commands show the estimation of a log-linear equation with a lagged dependent variable.

```SAMPLE 1 22
GENR LSAV=LOG(SAVINGS)
GENR LINC=LOG(INCOME)
* Estimate a log-linear model with a lagged dependent variable
GENR LSAVL1=LAG(LSAV)
SAMPLE 2 22
OLS LSAV LSAVL1 LINC / LOGLOG DLAG
DIAGNOS / CHOWONE=11
STOP
```

Note that the lagged dependent variable is included as the first explanatory variable. The `DLAG` option is used to obtain Durbin's h test as a test for autocorrelation when the model includes a lagged dependent variable.

The SHAZAM output can be viewed. The results from the `DIAGNOS` command are:

```SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS
N1   N2    SSE1        SSE2       CHOW    PVALUE    G-Q       DF1  DF2 PVALUE
11   10 0.13517     0.15749      1.8444    0.182  0.7510        8    7 0.346

CHOW TEST - F DISTRIBUTION WITH DF1=   3 AND DF2=  15
```

The Chow test statistic does not reject the null hypothesis of parameter stability and the Goldfeld-Quandt test statistic shows no evidence of heteroskedasticity. Durbin's h test statistic has the value 0.077 and so there is no evidence for autocorrelation in the errors. The conclusion is that the log-linear model with a lagged dependent variable reveals no evidence for a structural change during the Reagan-Bush presidency era. [SHAZAM Guide home]

#### SHAZAM commands for computing the Chow test statistic

The commands below show an example of programming in SHAZAM. The commands compute a Chow test statistic for the example given above. A p-value for the test statistic is also computed. The computations should replicate the Chow test statistic that is reported with the `DIAGNOS / CHOWTEST` command.

```SAMPLE 1 22
* Suppress output
SET NOOUTPUT
* OLS estimation for group 1
SAMPLE 1 11
OLS SAVINGS INCOME
* The sum of squared errors is available in the temporary variable \$SSE
GEN1 SSE1=\$SSE
GEN1 N1=\$N
* OLS estimation for group 2
SAMPLE 12 22
OLS SAVINGS INCOME
GEN1 SSE2=\$SSE
GEN1 N2=\$N
* OLS estimation for the complete sample.
SAMPLE 1 22
OLS SAVINGS INCOME
GEN1 SSEA=\$SSE
GEN1 K=\$K
* Compute the Chow test statistic
GEN1 SSEB=SSE1+SSE2
GEN1 DFDEN=N1+N2-2*K
GEN1 CHOW=((SSEA-SSEB)/K)/(SSEB/DFDEN)
* Get the p-value
SAMPLE 1 1
DISTRIB CHOW / TYPE=F DF1=K DF2=DFDEN CDF=CDF1
GEN1 PVAL=1-CDF1
PRINT CHOW PVAL
STOP
```

Note that following model estimation SHAZAM temporary variables are available with some results. These variables start with the \$ character. The above commands make use of the following temporary variables available after the `OLS` command.

 \$N The number of observations used in the OLS regression \$SSE The sum of squared errors \$K The number of coefficients [SHAZAM Guide home]

#### SHAZAM output

```|_SAMPLE 1 22
UNIT 88 IS NOW ASSIGNED TO: USECON.txt
3 VARIABLES AND       22 OBSERVATIONS STARTING AT OBS       1

|_* Estimate the savings-income relationship
|_OLS SAVINGS INCOME / RSTAT DWPVALUE

OLS ESTIMATION
22 OBSERVATIONS     DEPENDENT VARIABLE = SAVINGS
...NOTE..SAMPLE RANGE SET TO:    1,   22

DURBIN-WATSON STATISTIC  =   0.54879
DURBIN-WATSON P-VALUE =    0.000005

R-SQUARE =   0.6396     R-SQUARE ADJUSTED =   0.6216
VARIANCE OF THE ESTIMATE-SIGMA**2 =   997.69
STANDARD ERROR OF THE ESTIMATE-SIGMA =   31.586
SUM OF SQUARED ERRORS-SSE=   19954.
MEAN OF DEPENDENT VARIABLE =   136.91
LOG OF THE LIKELIHOOD FUNCTION = -106.128

VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
NAME    COEFFICIENT   ERROR      20 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
INCOME    0.31461E-01 0.5281E-02   5.958     0.000 0.800     0.7998     0.5790
CONSTANT   57.636      14.91       3.865     0.001 0.654     0.0000     0.4210

DURBIN-WATSON = 0.5488    VON NEUMANN RATIO = 0.5749    RHO =  0.70933
RESIDUAL SUM = -0.49738E-13  RESIDUAL VARIANCE =   997.69
SUM OF ABSOLUTE ERRORS=   536.24
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.6396
RUNS TEST:    5 RUNS,    9 POS,    0 ZERO,   13 NEG  NORMAL STATISTIC = -3.0039

|_DIAGNOS / CHOWONE=11

DEPENDENT VARIABLE = SAVINGS         22 OBSERVATIONS
REGRESSION COEFFICIENTS
0.314609350421E-01   57.6356858451

SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS
N1   N2    SSE1        SSE2       CHOW    PVALUE    G-Q       DF1  DF2 PVALUE
11   11  1010.8      5103.5      20.371    0.000  0.1981        9    9 0.012

CHOW TEST - F DISTRIBUTION WITH DF1=   2 AND DF2=  18

|_* Now consider a log-linear relationship
|_GENR LSAV=LOG(SAVINGS)
|_GENR LINC=LOG(INCOME)

|_OLS LSAV LINC / LOGLOG RSTAT DWPVALUE

OLS ESTIMATION
22 OBSERVATIONS     DEPENDENT VARIABLE = LSAV
...NOTE..SAMPLE RANGE SET TO:    1,   22

DURBIN-WATSON STATISTIC  =   0.67040
DURBIN-WATSON P-VALUE =    0.000040

R-SQUARE =   0.8095     R-SQUARE ADJUSTED =   0.8000
VARIANCE OF THE ESTIMATE-SIGMA**2 =  0.35331E-01
STANDARD ERROR OF THE ESTIMATE-SIGMA =  0.18797
SUM OF SQUARED ERRORS-SSE=  0.70663
MEAN OF DEPENDENT VARIABLE =   4.8416
LOG OF THE LIKELIHOOD FUNCTION(IF DEPVAR LOG) = -99.9099

VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
NAME    COEFFICIENT   ERROR      20 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
LINC      0.66122     0.7172E-01   9.219     0.000 0.900     0.8997     0.6612
CONSTANT -0.24110     0.5528     -0.4362     0.667-0.097     0.0000    -0.2411

DURBIN-WATSON = 0.6704    VON NEUMANN RATIO = 0.7023    RHO =  0.64948
RESIDUAL SUM =  0.19429E-15  RESIDUAL VARIANCE =  0.35331E-01
SUM OF ABSOLUTE ERRORS=   3.3446
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.8095
R-SQUARE BETWEEN ANTILOGS OBSERVED AND PREDICTED = 0.6859
RUNS TEST:    5 RUNS,   11 POS,    0 ZERO,   11 NEG  NORMAL STATISTIC = -3.0585

|_DIAGNOS / CHOWONE=11

DEPENDENT VARIABLE = LSAV            22 OBSERVATIONS
REGRESSION COEFFICIENTS
0.661217901001     -0.241102497959

SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS
N1   N2    SSE1        SSE2       CHOW    PVALUE    G-Q       DF1  DF2 PVALUE
11   11 0.11833     0.15911      13.923    0.000  0.7437        9    9 0.333

CHOW TEST - F DISTRIBUTION WITH DF1=   2 AND DF2=  18
|_STOP
``` [SHAZAM Guide home]

#### SHAZAM output

```|_SAMPLE 1 22
UNIT 88 IS NOW ASSIGNED TO: USECON.txt
3 VARIABLES AND       22 OBSERVATIONS STARTING AT OBS       1

|_GENR LSAV=LOG(SAVINGS)
|_GENR LINC=LOG(INCOME)
|_* Estimate a log-linear model with a lagged dependent variable
|_GENR LSAVL1=LAG(LSAV)
..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
|_SAMPLE 2 22
|_OLS LSAV LSAVL1 LINC / LOGLOG DLAG

OLS ESTIMATION
21 OBSERVATIONS     DEPENDENT VARIABLE = LSAV
...NOTE..SAMPLE RANGE SET TO:    2,   22

R-SQUARE =   0.8689     R-SQUARE ADJUSTED =   0.8543
VARIANCE OF THE ESTIMATE-SIGMA**2 =  0.22257E-01
STANDARD ERROR OF THE ESTIMATE-SIGMA =  0.14919
SUM OF SQUARED ERRORS-SSE=  0.40062
MEAN OF DEPENDENT VARIABLE =   4.8792
LOG OF THE LIKELIHOOD FUNCTION(IF DEPVAR LOG) = -90.6883

VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
NAME    COEFFICIENT   ERROR      18 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
LSAVL1    0.62422     0.1767       3.532     0.002 0.640     0.6673     0.6242
LINC      0.20643     0.1360       1.517     0.147 0.337     0.2867     0.2064
CONSTANT  0.27423     0.4841      0.5665     0.578 0.132     0.0000     0.2742

DURBIN-WATSON = 1.9737    VON NEUMANN RATIO = 2.0724    RHO =  0.00984
RESIDUAL SUM =  0.17847E-13  RESIDUAL VARIANCE =  0.22257E-01
SUM OF ABSOLUTE ERRORS=   2.2924
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.8689
R-SQUARE BETWEEN ANTILOGS OBSERVED AND PREDICTED = 0.8118
RUNS TEST:    8 RUNS,   11 POS,    0 ZERO,   10 NEG  NORMAL STATISTIC = -1.5603
DURBIN H STATISTIC (ASYMPTOTIC NORMAL) =  0.76882E-01

|_DIAGNOS / CHOWONE=11

DEPENDENT VARIABLE = LSAV            21 OBSERVATIONS
REGRESSION COEFFICIENTS
0.624221564206      0.206427954104      0.274228157820

SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS
N1   N2    SSE1        SSE2       CHOW    PVALUE    G-Q       DF1  DF2 PVALUE
11   10 0.13517     0.15749      1.8444    0.182  0.7510        8    7 0.346

CHOW TEST - F DISTRIBUTION WITH DF1=   3 AND DF2=  15
|_STOP
``` [SHAZAM Guide home]