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1 Nonlinear Regression Functions (SW Chapter 8)

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2 The TestScore STR relation looks linear (maybe)

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3 But the TestScore Income relation looks nonlinear...

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4 The general nonlinear population regression function

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6 Nonlinear Functions of a Single Independent Variable (SW Section 8.2)

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7 1. Polynomials in X

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8 Example: TestScore vs. Income

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9 Estimation in STATA

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10 Interpretation. graph twoway scatter testscr avginc || connected yhat avginc, sort msymbol(none) || connected yhat2 avginc, sort msymbol(none) See Chapter 3 of Statistics with Stata, especially pages 79 & 118

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11 Interpretation

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13 Interpretation

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14 Marginal effects in STATA

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15 Estimation of a cubic in STATA

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16 Plotting a cubic in STATA. graph twoway scatter testscr avginc || connected yhat2 avginc, sort msymbol(none) || connected yhat3 avginc, sort msymbol(T). reg testscr avginc avginc2 avginc3, rob. predict yhat3 (option xb assumed; fitted values)

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17 Marginal effects in STATA

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18 Marginal effects in STATA

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19 Ramseys RESET Test: REgression Specification Error Test Consider the model (1) General test for misspecification of functional form If LSA #1 holds, then no non-linear function of the Xs should be significant when added to the model. Consider (2) Null hypothesis is that (1) is correctly specified How many powers of predicted values to include? Conduct F-test on powers of predicted values J.B. Ramsey (1969), Tests for Specification Error in Classical Linear Least Squares Regression Analysis. Journal of the Royal Statistical Society, Series B 31, 350371

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20 Ramseys RESET Test. reg test str avginc, r Linear regression Number of obs = 420 F( 2, 417) = 132.65 Prob > F = 0.0000 R-squared = 0.5115 Root MSE = 13.349 ------------------------------------------------------------------------------ | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- str | -.6487401.3533403 -1.84 0.067 -1.34329.04581 avginc | 1.839112.114733 16.03 0.000 1.613585 2.064639 _cons | 638.7292 7.301234 87.48 0.000 624.3773 653.081 ------------------------------------------------------------------------------. estat ovtest (can just type. ovtest) Ramsey RESET test using powers of the fitted values of testscr Ho: model has no omitted variables F(3, 414) = 18.36 Prob > F = 0.0000

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21 Ramseys RESET Test. reg test str avginc avginc2, r Linear regression Number of obs = 420 F( 3, 416) = 286.55 Prob > F = 0.0000 R-squared = 0.5638 Root MSE = 12.629 ------------------------------------------------------------------------------ | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- str | -.9099512.3545374 -2.57 0.011 -1.606859 -.2130432 avginc | 3.881859.2709564 14.33 0.000 3.349245 4.414474 avginc2 | -.044157.0049606 -8.90 0.000 -.053908 -.034406 _cons | 625.2308 7.087793 88.21 0.000 611.2984 639.1631 ------------------------------------------------------------------------------. estat ovtest Ramsey RESET test using powers of the fitted values of testscr Ho: model has no omitted variables F(3, 413) = 2.48 Prob > F = 0.0605

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22 Ramseys RESET Test. reg test str avginc avginc2 avginc3, r Linear regression Number of obs = 420 F( 4, 415) = 207.23 Prob > F = 0.0000 R-squared = 0.5663 Root MSE = 12.608 ------------------------------------------------------------------------------ | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- str | -.9277523.3562919 -2.60 0.010 -1.628114 -.2273905 avginc | 5.124736.7045403 7.27 0.000 3.739824 6.509649 avginc2 | -.1011073.0287052 -3.52 0.000 -.157533 -.0446815 avginc3 |.0007293.0003414 2.14 0.033.0000582.0014003 _cons | 617.8974 7.926373 77.95 0.000 602.3165 633.4782 ------------------------------------------------------------------------------. estat ovtest Ramsey RESET test using powers of the fitted values of testscr Ho: model has no omitted variables F(3, 412) = 1.79 Prob > F = 0.1490

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23 Ramseys RESET Test. reg test str el_pct meal_pct, r Linear regression Number of obs = 420 F( 3, 416) = 453.48 Prob > F = 0.0000 R-squared = 0.7745 Root MSE = 9.0801 ------------------------------------------------------------------------------ | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- str | -.9983092.2700799 -3.70 0.000 -1.529201 -.4674178 el_pct | -.1215733.0328317 -3.70 0.000 -.18611 -.0570366 meal_pct | -.5473456.0241072 -22.70 0.000 -.5947328 -.4999583 _cons | 700.15 5.56845 125.74 0.000 689.2042 711.0958 ------------------------------------------------------------------------------. estat ovtest Ramsey RESET test using powers of the fitted values of testscr Ho: model has no omitted variables F(3, 413) = 6.29 Prob > F = 0.0004

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24 Ramseys RESET Test. reg test str el_pct meal_pct avginc, r Linear regression Number of obs = 420 F( 4, 415) = 467.42 Prob > F = 0.0000 R-squared = 0.8053 Root MSE = 8.4477 ------------------------------------------------------------------------------ | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- str | -.5603892.2550641 -2.20 0.029 -1.061768 -.0590105 el_pct | -.1943282.0332445 -5.85 0.000 -.2596768 -.1289795 meal_pct | -.3963661.0302302 -13.11 0.000 -.4557895 -.3369427 avginc |.674984.0837161 8.06 0.000.5104236.8395444 _cons | 675.6082 6.201865 108.94 0.000 663.4172 687.7992 ------------------------------------------------------------------------------. estat ovtest Ramsey RESET test using powers of the fitted values of testscr Ho: model has no omitted variables F(3, 412) = 0.47 Prob > F = 0.7014

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25 Ramseys RESET Test: replicated. predict yh (option xb assumed; fitted values). sum yh Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- yh | 420 654.1565 17.09817 614.9183 702.8387. gen yhz = (yh-r(mean))/r(sd). sum yh* Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- yh | 420 654.1565 17.09817 614.9183 702.8387 yhz | 420 1.22e-09 1 -2.294882 2.847214. gen yhz2=yhz*yhz. gen yhz3=yhz^3. gen yhz4=yhz^4

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26 Ramseys RESET Test: replicated. reg test str el meal avginc yhz2 yhz3 yhz4 Source | SS df MS Number of obs = 420 -------------+------------------------------ F( 7, 412) = 244.48 Model | 122595.145 7 17513.5921 Prob > F = 0.0000 Residual | 29514.4488 412 71.6370116 R-squared = 0.8060 -------------+------------------------------ Adj R-squared = 0.8027 Total | 152109.594 419 363.030056 Root MSE = 8.4639 ------------------------------------------------------------------------------ testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- str | -.5500585.2336368 -2.35 0.019 -1.009327 -.0907896 el_pct | -.2170374.0407058 -5.33 0.000 -.2970544 -.1370204 meal_pct | -.400967.0289303 -13.86 0.000 -.4578364 -.3440976 avginc |.6476592.1505253 4.30 0.000.3517657.9435527 yhz2 |.7652051.915534 0.84 0.404 -1.034495 2.564906 yhz3 | -.0822669.3243362 -0.25 0.800 -.7198272.5552933 yhz4 | -.0650369.1767693 -0.37 0.713 -.412519.2824453 _cons | 675.8077 5.443279 124.15 0.000 665.1076 686.5077 ------------------------------------------------------------------------------. test yhz2 yhz3 yhz4 ( 1) yhz2 = 0 ( 2) yhz3 = 0 ( 3) yhz4 = 0 F( 3, 412) = 0.47 Prob > F = 0.7014

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27 2. Logarithmic functions of Y and/or X

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28 The 3 log specifications

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29 I. Linear-log population regression function

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30 Linear-log case, continued

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31 Example: TestScore vs. ln(Income)

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32 Linear-log vs. Cubic models

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33 II. Log-linear population regression function

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34 Log-linear case, continued

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35 III. Log-log population regression function

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36 Log-log case, continued

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37 Example: ln( TestScore) vs. ln( Income)

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38 Example: ln( TestScore) vs. ln( Income), ctd.

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39 The log-linear and log-log specifications:

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45 Other nonlinear functions (and nonlinear least squares) (SW App. 8.1)

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46 Negative exponential growth

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47 Nonlinear Least Squares

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