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PPA 207: Quantitative Methods Meeting 10, Spring 2004 1. Homework a. Studenmund, Chapter 8, Number 9 a. I expect that competitor's price (PC), gross domestic product (Y), consumption (C), and number of stores (N) will have a positive influence on sales of durable goods (SQ). I expect that own price (PQ) will have a negative influence on SQ. There are 26 observations, so dof equals 26-5-1=20. These are all one-tailed tests so critical t at 5 percent level is 1.725. The calculated t for PC is 0.80. for PQ it is 1.20, for Y it is 0.51, for C it is 1.49, for N it is 1.94; therefore, can only "accept" that N has a positive influence on SQ. b. Omitted variables could be measure of differences in advertising activity or product mix across the years. Y and C are really measuring the same thing (the effect of national income on purchases at these stores) and hence they are likely to be highly collinear and one of them is irrelevant. c. The high R squared and the high colinearity between Y and C indicates that multicolinearity is likely to be a problem. The partial correlation coefficient between of 0.813 between PC and PQ is not that large and these two variables measure two separate influences on SQ (the own price effect and the substitute price effect). Therefore I would not consider dropping one of these variables. d. I would recommend removing Y and keeping C as a more direct measure of what influences SQ. Also look to add some explanatory variables to account for marketing and product differences over the years. b. Pollock, Chapter 7, Question 2 Review answers in class c. Go over one student’s regression run and multicolinearity check and correction 2. Studenmund, Chapter 9, Serial Correlation Consider that you have been asked to build a simple model of property tax revenue in Sacramento County in year “t” Property tax revenuet = f (populationt, timet) Either for understanding impact of these causal variables or forecasting Data retrieved from California Institute for County Government Web Site COUNTY SACRAMENTO SACRAMENTO SACRAMENTO SACRAMENTO SACRAMENTO SACRAMENTO SACRAMENTO SACRAMENTO SACRAMENTO SACRAMENTO SACRAMENTO SACRAMENTO SACRAMENTO SACRAMENTO YEAR 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 PROPTXRV 86910539 99877113 113703377 127981910 137191499 154891300 173025724 186340243 181929930 129401948 98928752 104472678 104628365 106072967 POP 891113 916044 948523 979279 1.01E+06 1.05E+06 1.08E+06 1.09E+06 1.09E+06 1.09E+06 1.10E+06 1.12E+06 1.13E+06 1.18E+06 Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Run a time series regression Likely problem to arise: serial correlation Violation of classical assumption IV Observations on the error term are likely to be correlated First-order serial correlation (most common) et = ρet-1 + ut ρ = first-order autocorrelation coefficient -1 < ρ < +1 Positive serial correlation (ρ > 0) Most likely to observe Unaccounted for shock in the economy raises residual value in period “t” Causes residual value to be higher in periods “t+1”, “t+2”, etc. See Figure 9.1 Shocks at peaks and troughs of residual graphs Impure serial correlation Caused by specification error Omitted variable or an incorrect functional form Problems created by this are picked up in error terms So first correction is to deal with this Consequences of serial correlation (1) Does not bias the regression coefficient estimates See Figure 9.6 Distribution of regression coefficient estimates are still centered around true value (2) Increases the variance of the regression coefficient estimates See Figure 9.6 (3) Causes OLS to underestimate the standard errors attached to regression coefficients Usually allows a better fit than non-serially correlated observations would allow Everything is moving together Values calculated for t statistics, F statistic, and R-squared are too high Consequence of serial correlation Regression to predict real interest rate in year t Detection of serial correlation: Durbin-Watson d statistic Examine the residuals of a suspect regression Run property tax revenue in SPSS Click on Durbin-Watson statistic and retrieve unstandardized residuals Graph residuals against time to look for pattern Model Summary(b) Model 1 R R Square .911(a) .829 Adjusted R Square .798 Std. Error of the Estimate Durbin-Watson 14879529.7534 1.487 a Predictors: (Constant), time, pop b Dependent Variable: proptaxrv Coefficients(a) Unstandardized Coefficients Model 1 (Constant) pop time a Dependent Variable: proptaxrv B Std. Error 1062219011. 360 163188628.8 04 1322.577 26004688.61 3 Standardized Coefficients Beta t Sig. -6.509 .000 181.127 3.429 7.302 .000 3715879.302 -3.286 -6.998 .000 Unstandardized Residual 30000000.00000 20000000.00000 10000000.00000 0.00000 -10000000.00000 -20000000.00000 2.50 5.00 7.50 10.00 12.50 time DW statistic equals 1.487 Sample size = 14, one constant, and two explanatory variables estimated (k = 3) Two-sided 90% test of the null hypothesis of no serial correlation See Table B-4 dL = 0.81 and du = 1.75 See Figure 9.7 DW statistic falls in inconclusive range First try correcting functional form Try a non-linear form Log of property tax revenue and population Result better because DW statistic larger and closer to acceptance of no serial correlation Model Summary(b) Model 1 R R Square .932(a) .868 Adjusted R Square Std. Error of the Estimate .844 Durbin-Watson .09786 1.567 a Predictors: (Constant), time, lpop b Dependent Variable: lproptaxrv Coefficients(a) Unstandardized Coefficients Model 1 B (Constant) Std. Error -112.232 15.382 lpop 9.542 1.122 time -.182 .023 a Dependent Variable: lproptaxrv Standardized Coefficients Beta t Sig. -7.296 .000 3.233 8.507 .000 -3.066 -8.069 .000 Try adding omitted variable(s) Look at data for clear break points (here 1994 – ERAF) Create an ERAF dummy for years 1994 and beyond Remove time dummy Results Better, but still not in acceptance range Model Summary(b) Model 1 R Adjusted R Square R Square .930(a) .864 Std. Error of the Estimate .840 Durbin-Watson .09928 1.582 a Predictors: (Constant), Eraf, lpop b Dependent Variable: lproptaxrv Coefficients(a) Unstandardized Coefficients Model 1 B (Constant) Standardized Coefficients Std. Error -25.760 6.097 lpop 3.219 .441 Eraf -.591 .075 Beta t Sig. -4.225 .001 1.091 7.296 .000 -1.186 -7.934 .000 a Dependent Variable: lproptaxrv Try adding omitted variable(s) Look at data for clear break points (here 1994 – ERAF) Create an ERAF dummy for years 1994 and beyond Remove time dummy Results Better, but still not in acceptance range Could correct with Cochrane-Orcutt method Done directly in SPSS Analyze=>time series=>autoregression First produces an estimate of ρ (here = 0.65) Second runs new regression 9.18 on p. 330 Results are so poor, I would not use FINAL PARAMETERS: Estimate of Autocorrelation Coefficient Rho Standard Error of Rho .65453521 .23907816 Cochrane-Orcutt Estimates Multiple R R-Squared .81154849 .65861096 Adjusted R-Squared Standard Error Durbin-Watson .54481461 .09406975 1.2995309 Variables in the Equation: B SEB BETA T .8230470 1.235202 .13754041 .6663260 -.4008094 .097717 -.84666687 -4.1017485 7.4652984 17.167155 SIG T lpop .52191470 eraf .00266972 CONSTANT .67390521 . .4348594 3. Check Learning PowerPoint Web Link Review over Spring Break if weak in this area Presentation required at last meeting in regard to your Paper 4. Homework Due the Start of Meeting 11 (April 13) (1) Would anyone like to prepare a 10 minute presentation on Kahn article for April 13 that would highlight his theoretical model, his specification (linear or log), his corrections of multicolinearity/heteroscedasticity, and an interpretation of his regression results? You will have access to a projection of article in classroom. Just prepare some overheads or notes to lead a discussion on this topic. Grade granted will substitute for one HW grade. More than one person can do. (2) Read all of the material under meeting eleven in the syllabus; come prepared to discuss. (3) A typed and well developed question from reading assignment for week eleven. (4) Answer question 11 in Studemund, Chapter 9. (5) By Friday (April 2) I will have updated the Checklist for Final Paper at http://www.csus.edu/indiv/w/wassmerr/ppa207pa.htm and a very good example of a student paper at http://www.csus.edu/indiv/w/wassmerr/paperdire.pdf . Download both of these after then and spend some time over break looking over, we will discuss upon return. (6) The final paper that you will write must contain a description of at least three other pieces of academic research in the area. You can find this research by searching at the CSUS Library’s web page of literature bases that must be accessed on campus or using a SacLink account (http://library.csus.edu/databases/. I would suggest using ECONLIT and EBSCOhost as two literature sources that will have regression studies in them. Search using keywords that include "regression" and your topic. Be sure that 2/3 of the articles you choose use some form of regression analysis. Find your three articles over the break and turn in next time a Xeroxed copy of the front page (including abstract) of each article. (You will need complete copies of articles for yourself.)