11 - James Madison University College of Business



Answers Chapter 11

11.1, 11.2, 11.7, 11.10 (parts a and b only).

1. See class notes on April 19th …we did all of these transformations in class.

|Specification |Transformation: |Why??? |

|for var(et) | | |

|[pic] |Divide the model by X1/4: independent |We divide by the standard deviation, which is|

| |variables: 1/X1/4 and X/X1/4 |the square root of the variance : X1/4. |

| | |We can ignore the ( term in all of the models|

| | |since it doesn’t vary over observations. |

| | | |

|(2xt |Divide the model by X1/2 |This is just like the one we did in class. |

| |Independent variables: 1/X1/2 and X/X1/2 |The standard deviation is [pic] |

| | | |

|[pic] |Divide the model by X: independent variables:|Here, the standard deviation is xt, so we |

| |1/x plus an intercept |divide by xt |

|(2ln(xt) |Divide the model by (ln(X))1/2 |Here the standard deviation is |

| |Independent variables: 1/(ln(x))1/2 |[pic] |

Here is the SAS code and output from estimating the food expenditure model for each of these transformations. Make sure to look through the results and pick out the estimates of (1 and (2.

options ls=78;

options formdlim='*';

data food;

infile 'c:\documents and settings\doylejm\my documents\classes\ec385\ue\datafiles\table3-1.dat'

firstobs=2;

input y x;

proc reg ;

white: model y = x / acov;

run;

data foodgls;

set food;

y_a = y/(x**(1/4));

x1_a = 1/(x**(1/4));

x2_a = x/(x**(1/4));

y_b = y/sqrt(x);

x1_b = 1/sqrt(x);

x2_b = x/sqrt(x);

y_c = y/x;

x_c = 1/x;

lnx = log(x);

y_d = y/sqrt(lnx);

x1_d = 1/sqrt(lnx);

x2_d = x/sqrt(lnx);

proc reg;

Part_A: model y_a = x1_a x2_a / noint;

Part_B: model y_b = x1_b x2_b / noint;

Part_C: model y_c = x_c;

Part_D: model y_d = x1_d x2_d / noint;

run;

Model: white

Dependent Variable: y

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 1 25221 25221 17.65 0.0002

Error 38 54311 1429.24556

Corrected Total 39 79533

Root MSE 37.80536 R-Square 0.3171

Dependent Mean 130.31300 Adj R-Sq 0.2991

Coeff Var 29.01120

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 40.76756 22.13865 1.84 0.0734

x 1 0.12829 0.03054 4.20 0.0002

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

The SAS System 10

The REG Procedure

Model: white

Dependent Variable: y

Consistent Covariance of Estimates

Variable Intercept x

Intercept 561.88951794 -0.886643258

x -0.886643258 0.0014569032

Below are 4 sets of regression estimates, for parts a) – d). The estimates that are highlighted in yellow are estimates of the (1 (intercept in the original model) and those highlighted blue are estimates of the (2 (slope in the original model).

Model: Part_A

Dependent Variable: y_a

NOTE: No intercept in model. R-Square is redefined.

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 2 26028 13014 258.20 |t|

x1_a 1 36.75257 20.05232 1.83 0.0747

x2_a 1 0.13391 0.02879 4.65 F

Model 2 978.84644 489.42322 270.71 |t|

x1_b 1 31.92438 17.98608 1.77 0.0839

x2_b 1 0.14096 0.02700 5.22 F

Model 1 0.00562 0.00562 2.30 0.1377

Error 38 0.09285 0.00244

Corrected Total 39 0.09846

Root MSE 0.04943 R-Square 0.0571

Dependent Mean 0.19116 Adj R-Sq 0.0322

Coeff Var 25.85840

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 0.15769 0.02342 6.73 F

Model 2 106685 53343 249.90 |t|

x1_d 1 39.55015 21.46901 1.84 0.0733

x2_d 1 0.12996 0.02997 4.34 0.0001

11.2 Code and SAS output appear below.

(a) Countries with high per capita income can decide whether to spend larger amounts on education than their poorer neighbours, or to spend more of their larger income on other things. They are likely to have more discretion with respect to where public monies are spent. On the other hand, countries with low per capita income may regard a particular level of education spending as essential, meaning that they have less scope for deviating from a mean function. These differences can be captured by a model with heteroskedasticity. Remember that heteroskedasticity is more common in cross-section data.

(b) The least squares estimated function is

[pic] [pic]

(0.0485) (0.00518)

This function and the corresponding residuals appear in Figure 11.1. The absolute magnitude of the errors does tend to increase as x increases suggesting the existence of heteroskedasticity.

[pic]

(c) Since it is suspected that, if heteroskedasticity exists, the variance is related to xt , we begin by ordering the observations according to the magnitude of xt. Then, splitting the sample into two equal subsets of 17 observations each, and applying least squares to each subset, we obtain [pic] = 0.0081608 and [pic] = 0.029127 leading to a Goldfelt-Quandt statistic of

[pic] = 3.569

The critical value from an F-distribution with (15,15) degrees of freedom and a 5% significance level is Fc = 2.40. Since 3.569 > 2.40 we reject a null hypothesis of homoskedasticity and conclude that the error variance is directly related to per capita income xt.

(e) Generalized least squares estimation under the assumption [pic] yields

[pic]

(0.0289) (0.00441)

(note: I have expressed these results in the model’s original form although it was estimated with no intercept and two independent variables: the reciprocal of the square root of x and x over the square root of x.) The estimated response of per capita education expenditure to per capita income has declined slightly relative to the least squares estimate. The associated 95% confidence interval is (0.0603, 0.0783). This interval is narrower than both those computed from least squares estimates. The comparison with the White-calculated interval suggests that generalized least squares is more efficient; a comparison with the conventional least squares interval is not really valid because the standard errors used to compute that interval are not valid. See below for the case were Var(et) = (2X2t. The differences of how this is carried out and how to interpret the results is important.

Part B Least Squares results

The REG Procedure

Model: MODEL1

Dependent Variable: y

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 1 3.68386 3.68386 199.59 |t|

Intercept 1 -0.12457 0.04852 -2.57 0.0151

x 1 0.07317 0.00518 14.13 F

Model 1 0.42220 0.42220 14.50 0.0017

Error 15 0.43690 0.02913

Corrected Total 16 0.85910

Root MSE 0.17067 R-Square 0.4914

Dependent Mean 0.78115 Adj R-Sq 0.4575

Coeff Var 21.84803

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 -0.14087 0.24569 -0.57 0.5749

x 1 0.07516 0.01974 3.81 0.0017

This regression gets you the denominator for the GQ-statistic

The REG Procedure

Model: MODEL1

Dependent Variable: y

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 1 0.14225 0.14225 17.43 0.0008

Error 15 0.12241 0.00816

Corrected Total 16 0.26466

Root MSE 0.09034 R-Square 0.5375

Dependent Mean 0.17232 Adj R-Sq 0.5066

Coeff Var 52.42382

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 -0.03807 0.05495 -0.69 0.4990

x 1 0.05047 0.01209 4.17 0.0008

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

These are the critical values The SAS System 17

Obs fc tc

1 2.40345 2.03693

This regression corrects for heteroskedasticity of the form var(et) = (2Xt

The REG Procedure

Model: MODEL1

Dependent Variable: ystar

NOTE: No intercept in model. R-Square is redefined.

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 2 0.96083 0.48041 242.45 |t|

x1star 1 -0.09292 0.02890 -3.21 0.0030

x2star 1 0.06932 0.00441 15.71 F

Model 1 0.00349 0.00349 12.69 0.0012

Error 32 0.00880 0.00027504

Corrected Total 33 0.01229

Root MSE 0.01658 R-Square 0.2840

Dependent Mean 0.05153 Adj R-Sq 0.2616

Coeff Var 32.18259

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 0.06443 0.00460 13.99 Fc = 2.33, we reject H0 and conclude that heteroskedasticity exists, and is dependent on xt.

(ii) When we order the observations with respect to nt , there is not a unique ordering because nt takes on repeated integer values. There are 8 observations where nt = 3. One of these values must be included in the first 19 observations, the other 7 in the last 19 observations. There are 8 ways of doing this. The results from SAS, EViews and SHAZAM are as follows.

[pic] (SAS)

These values are greater than 2.33, and so we reject a null hypothesis of homoskedasticity and conclude that the error variances are dependent on nt. These test outcomes are consistent with the evidence provided by the residual plots in part (b).

(d) The alternative variance estimators yield the following standard errors:

| |Standard Errors |

|Coefficients |White |Least Squares |

|(2 |0.0287 |0.0354 |

|(3 |0.4360 |0.4140 |

The results from White's variance estimator suggest the usual least squares results would underestimate the reliability of estimation for (2 and overestimate the reliability of estimation for (3.

(e) To find generalized least squares estimates when [pic] we begin by calculating ht for each observation. Then we apply least squares to the transformed model.

[pic]

The resulting estimates, with those from least squares, and the White standard errors are in the table below. The two estimates for (2 are similar, but the GLS estimate for the response of food expenditure to an additional household member is noticeably higher. The standard errors suggest that (1 and (3 have been more precisely estimated by GLS, but not (2. However, we do need to keep in mind that standard errors are square roots of estimated variances. It is possible for an improvement in precision to take place even when it is not reflected by the standard errors.

|Variable |GLS |LS (White) |

|constant |1.682 |2.243 |

| |(1.760) |(2.270) |

|xt |0.160 |0.165 |

| |(0.032) |(0.029) |

|nt |1.364 |1.145 |

| |(0.285) |(0.436) |

11.10 (a) The graphs for plotting the residuals against income and age show that the absolute values of the residuals increase as income increases but they appear to be constant as age increases. This indicates that the error variance depends on income.

(b) Since the residual plot shows that the error variance may increase when income increases, and this is a reasonable outcome since greater income implies greater flexibility in travel, we set the null and alternative hypotheses as [pic]. The test statistic is

[pic]

The 5% critical value for (96, 96) degrees of freedom is [pic]. Thus, we reject [pic] and conclude that the error variance depends on income.

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