2SLS HATCO SPSS and SHAZAM Example. by Eddie Oczkowski. August X9: Usage Level (how much of the firm s total product is purchased from HATCO).

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1 2SLS HATCO SPSS and SHAZAM Example by Eddie Oczkowski August 200 This example illustrates how to use SPSS to estimate and evaluate a 2SLS latent variable model. The bulk of the example relates to SPSS, the SHAZAM code is provided on the final page. We employ data from Hair et al (Multivariate Data Analysis, 998). The data pertain to a company called HATCO and relate to purchase outcomes from and perceptions of the company. The models presented may not necessarily be good models, we simply use them for presentation purposes. Consider a model which has a single dependent variable (usage) and two latent independent variables (strategy and image). Dependent variable X9: Usage Level (how much of the firm s total product is purchased from HATCO). Latent Independent Variables Strategy X: Delivery Speed (assume this is the scaling variable) X2: Price Level X3: Price Flexibility X7: Product Quality Image X4: Manufacturer s Image (assume this is the scaling variable) X6: Salesforce image

2 2SLS Estimation The 2SLS option is gained via: Analyze Regression 2-Stage Least Squares For our basic model (usage against strategy and image) the variable boxes are filled by: Dependent Variable: Explanatory Variables: Instrumental Variables: X9 X and X4 (these are our scaling variables) X2, X3, X7 and X6 (these are our non-scaling variables) 2

3 For the diagnostic testing of the model it is useful to save the residuals and predictions from this model using Options. Part of the output from this 2SLS model is: Two-stage Least Squares Equation number: Dependent variable.. X9 Multiple R R Square Adjusted R Square Standard Error Analysis of Variance: DF Sum of Squares Mean Square Regression Residuals F = Signif F =

4 Variables in the Equation Variable B SE B Beta T Sig T X X (Constant) The following new variables are being created: Name Label FIT_ Fit for X9 from 2SLS, MOD_2 Equation ERR_ Error for X9 from 2SLS, MOD_2 Equation Comments: The R-Square is 0.34 and F-statistic being significant indicates reasonable overall fit. The two independent variables are both statistically significant with expected positive signs. Two variables have been created: FIT_ is the IV fitted value variable while ERR_ is the IV residual. 2SLS as two OLS Regressions Consider now the 2 step method for calculating estimates. This should be employed to get the 2SLS forecasts and residuals for later diagnostic testing. The first step is to run a regression for each scaling variable against all instruments and save predictions. OLS Regression: X against X2, X3, X6, X7, save predictions. OLS Regression: X4 against X2, X3, X6, X7, save predictions. Recall the R-square values from these runs can be examined to ascertain the possible usefulness of the instruments. 4

5 The standard OLS option is gained via: Analyze Regression Linear The st regression is: OLS Regression: X against X2, X3, X6, X7, save predictions. 5

6 Save the predictions in the Save box. Part of the output from the regression is: Regression Model Model Summary b Adjusted Std. Error of R R Square R Square the Estimate.604 a a. Predictors: (Constant), Product Quality, Salesforce Image, Price Flexibility, Price Level b. Dependent Variable: Delivery Speed 6

7 Model (Constant) Price Level Price Flexibility Salesforce Image Product Quality Coefficients a Unstandardized Coefficients a. Dependent Variable: Delivery Speed Standardi zed Coefficien ts B Std. Error Beta t Sig E Comments: The R-square exceeds 0.0 and some variables are significant, this indicates some instrument acceptability. Note, however, that Price Level appears not to be a good instrument. A new variable with the predictions has been saved here: pre_. The same approach is used for the other scaling variable. OLS Regression: X4 against X2, X3, X6, X7, save predictions. Part of the output from this regression is: Regression Model Model Summary b Adjusted Std. Error of R R Square R Square the Estimate.799 a a. Predictors: (Constant), Product Quality, Salesforce Image, Price Flexibility, Price Level b. Dependent Variable: Manufacturer Image 7

8 Model (Constant) Price Level Price Flexibility Salesforce Image Product Quality Coefficients a Unstandardized Coefficients a. Dependent Variable: Manufacturer Image Standardi zed Coefficien ts B Std. Error Beta t Sig E E Comments: The R-square is much better here, and so the instruments appear to be better for image rather than strategy. Here clearly Salesforce Image is the key instrument for the image scaling variable. A new variable with the predictions has been saved here: pre_2. The final step in the process is to OLS regress the dependent variable (X9) on the two new prediction variables (pre_ and pre_2). 8

9 To produce the 2SLS forecasts and residuals we need to use the Save option: Part of the output from the 2 nd stage regression is: Regression Model Model Summary b Adjusted Std. Error of R R Square R Square the Estimate.530 a a. Predictors: (Constant), Unstandardized Predicted Value, Unstandardized Predicted Value b. Dependent Variable: Usage Level 9

10 Model Regression Residual Total ANOVA b Sum of Squares df Mean Square F Sig a a. Predictors: (Constant), Unstandardized Predicted Value, Unstandardized Predicted Value b. Dependent Variable: Usage Level Model (Constant) Unstandardized Predicted Value Unstandardized Predicted Value a. Dependent Variable: Usage Level Coefficients a Unstandardized Coefficients Standardi zed Coefficien ts B Std. Error Beta t Sig Comments: Note how the parameter estimates are the same between this regression and the initial 2SLS model. Also note how the standard errors (and hence t and significance 2 levels) are different. The reported R-square is the ( GR ) generalized R-square referred to in the notes and this indicates how 28.% of the variation in the data is explained. This is different to the initially presented R-square in the 2SLS model of 34.6%. Two new variables have been saved: pre_3 which are the 2SLS forecasts and res_ which are the 2SLS residuals. Over-identifying Restrictions Test To perform this test we perform a regression of the IV residuals (err_) against all the instruments: X2, X3, X6, X7. Note the R-square from this regression and multiply it by the sample size (N = 00) to get the test statistic. In this case the degrees of freedom (no. of instruments less no. of RHS variables) is (4 2 = 2). At the 5% level of significance the critical value for a chi-square with d.f. = 2 is:

11 The relevant regression window is: Part of the output from this regression is: Regression Model Model Summary Adjusted Std. Error of R R Square R Square the Estimate.680 a a. Predictors: (Constant), Product Quality, Salesforce Image, Price Flexibility, Price Level

12 Model (Constant) Price Level Price Flexibility Salesforce Image Product Quality Coefficients a Unstandardized Coefficients Standardi zed Coefficien ts B Std. Error Beta t Sig a. Dependent Variable: Error for X9 from 2SLS, MOD_2 Equation Comments: The R-square is and so the test statistic is: N * R-Square = 00 (0.462) = 46.2, this far exceeds the critical value of 5.99 and therefore we conclude that there is a model specification problem or the instruments are invalid. There is a major problem here. Note, all the instruments are significant in this equation illustrating how the instruments can explain significant amounts of the variation in the residuals. RESET (Specification Error Test) To perform this test we first need to compute the square of the 2SLS forecasts. That is we need to compute: pre_3 *pre_3. We can call the new variable whatever we want, say, pre_32. 2

13 To do this we use the option: Transform Compute The new variable pre_32 is now added to the original 2SLS model. That is, we employ the original dependent, independent and instrumental variables, but we add to the independent variables and instrumental variables pre_32. Part of the output from this 2SLS regression is: Two-stage Least Squares Dependent variable.. X9 Multiple R R Square Adjusted R Square.2483 Standard Error

14 Variables in the Equation Variable B SE B Beta T Sig T X X PRE_ (Constant) Comments: The test statistic is the t-ratio for pre_32. In this case the t-ratio is with a p-value of This is highly insignificant. This implies that there are no omitted variables and the functional form can be trusted. Taken together with the previous test, this may imply that the problems with the model relate to inadequate instruments. Heteroscedasticity Test To perform this test we initially have to square the IV residuals using the compute option: err_2 = err_ * err_ 4

15 This new variable (err_2) is then regressed against the 2SLS forecasts (pre_32) and the t-ratio on the forecast variable represents the test statistic. The output from this regression is: Regression Model Model Summary Adjusted Std. Error of R R Square R Square the Estimate.069 a a. Predictors: (Constant), PRE_32 5

16 Model (Constant) PRE_32 Unstandardized Coefficients a. Dependent Variable: ERR_2 Coefficients a Standardi zed Coefficien ts B Std. Error Beta t Sig E The t-ratio on pre_32 is with a p-value of 0.496, this is highly insignificant indicating the absence of heteroscedastcity. Interaction Effects To illustrate interaction effects, assume that strategy and image interact to create a new interaction latent independent variable. This variable is in addition to the original two independent variables. To create the new variables we employ the transform compute option. For the new independent variable we multiply the scaling variables by each other: say XX4 = X*X4 6

17 The instruments for this new variable are the products of all the remaining non-scaling variables across the two constructs. Since there is only one non-scaling variable for image we simply multiply it with the non-scaling variables for strategy to get our instruments: X2X6 =X2*X6 X3X6 = X3*X6 X7X6 = X7*X6 Thus the original 2SLS model is run again with one new explanatory variable XX4 and three new instrumental variables X2X6, X3X6, X7X6. Part of the output from this 2SLS regression is: Two-stage Least Squares Dependent variable.. X9 Multiple R R Square.3486 Adjusted R Square Standard Error

18 Variables in the Equation Variable B SE B Beta T Sig T X X XX (Constant) Comments: Note, this model appears to be inferior to the original specification. All the variables are now insignificant, including the new interaction term XX4. Non-nested Testing To illustrate these tests consider two models: Model A: Model B: Usage Strategy Usage Image Assume we wish to ascertain which variable better explains usage. We will conduct a paired test alternating the role of Models A and B. Case H0: Null model: Usage Strategy H: Alternative model: Usage Image In terms of our notation, our x s are the strategy indicators while the w s are the image indicators. The three steps are:. Regression: X4 on X6 and save the predictions (pre_4). 2. 2SLS regression X9 on X and pre_4 (instruments: X2, X3, X7 and pre_4). 3. The t-ratio on the pre_4 variable is the test statistic. The output from this 2SLS regression is: Two-stage Least Squares Dependent variable.. X9 Multiple R R Square.3445 Adjusted R Square Standard Error

19 Variables in the Equation Variable B SE B Beta T Sig T X PRE_ (Constant) Comments: The t-ratio for Pre_4 is 2.77 with a p-value of , this is highly significant. This implies that the alternative model H image rejects the null model H0 strategy. Case 2 H0: Null model: Usage Image H: Alternative model: Usage Strategy In terms of our notation our, x s are the image indicators while the w s are the strategy indicators. The three steps are:. Regression: X on X2,X3,X7 and save the predictions (pre_5). 4. 2SLS regression X9 on X4 and pre_5 (instruments: X6 and pre_5). 5. The t-ratio on the pre_5 variable is the test statistic. The output from this 2SLS regression is: Two-stage Least Squares Dependent variable.. X9 Multiple R R Square Adjusted R Square Standard Error Variables in the Equation Variable B SE B Beta T Sig T X PRE_ (Constant) Comments: The t-ratio for Pre_5 is with a p-value of , this is highly significant. This implies that the alternative model H strategy rejects the null model H0 image. In summary these results combined imply that both models reject each other and therefore it is erroneous to use either in isolation. 9

20 2SLS HATCO SHAZAM EXAMPLE This section presents the SHAZAM code corresponding to the SPSS example. * Original 2SLS model 2SLS X9 X X4 (X2 X3 X7 X6) / PREDICT=FIT_ RESID=ERR_ * 2 step OLS version to get 2SLS predictions, residuals and GR^2 OLS X X2 X3 X6 X7 / PREDICT=PRE_ OLS X4 X2 X3 X6 X7 / PREDICT=PRE_2 OLS X9 PRE_ PRE_2 / PREDICT=PRE_3 RESID=RES_ * Over-identifying restrictions test OLS ERR_ X2 X3 X6 X7 *RESET test GENR PRE_32=PRE_3*PRE_3 2SLS X9 X X4 PRE_32 (X2 X3 X7 X6 PRE_32) * Heteroscedasticity Test GENR ERR_2=ERR_*ERR_ OLS ERR_2 PRE_32 * Interactions Model Specification GENR XX4=X*X4 GENR X2X6=X2*X6 GENR X3X6=X3*X6 GENR X7X6=X7*X6 2SLS X9 X X4 XX4 (X2 X3 X7 X6 X2X6 X3X6 X7X6) * Non-nested Test Case OLS X4 X6 / PREDICT=PRE_4 2SLS X9 X PRE_4 (X2 X3 X7 PRE_4) * Non-nested Test Case 2 OLS X X2 X3 X7 / PREDICT=PRE_5 2SLS X9 X4 PRE_5 (X6 PRE_5) 20

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