ans_hw4_eco321_2010f

ans_hw4_eco321_2010f - Economic Statistics II Solutions -...

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Unformatted text preview: Economic Statistics II Solutions - Homework Assignment 4 Use the data set usmacro.dta that contains the monthly US unemployment rate [%]: Stata's variable name unemp, from 1984m1 to 2010m3. Define U nempt to be the unemployment rate. Assume that the error term is homoskedastic. (a) Construct a graph of the historical data of the unemployment rate. FIGURE-1: Unemployment Rate 12 10 unemp 8 6 4 1985m1 1993m5 time 2001m9 2010m1 (b) Find the 1st- to 10th-autocorrelation of U nempt . TABLE 1: Autocorrelations Lag 1 2 3 4 5 6 7 8 9 10 U nemp .0593 -.1555 -.3307 -.0340 .2207 .1107 .2164 -.0444 -.3693 -.1720 1 (c) Run the regression of AR(1), AR(2), AR(3), and AR(4) model for U nempt . TABLE 2: Autoregressions of U nempt AR(1) U nempt-1 U nempt-2 U nempt-3 U nempt-4 .9687 (.0160) AR(2) 1.0471 (.0565) -.0809 (.0572) AR(3) 1.0569 (.0564) -.2245 (.0815) .1432 (.0571) AR(4) 1.0098 (.0540) -.1531 (.0782) -.1861 (.0788) .3233 (.0549) .0455 (.0967) .9289 311 37.843 const. R T SSR 2 .1878 (.0964) .9209 314 43.642 .2031 (.0984) .9205 313 43.264 .1494 (.1001) .9211 312 42.363 (d) Run the regression of AR(1), AR(2), AR(3), and AR(4) model for U nempt. TABLE 3: Autoregressions of U nempt AR(1) U nempt-1 U nempt-2 U nempt-3 U nempt-4 .0594 (.0160) AR(2) .0661 (.0565) -.1597 (.0572) AR(3) .0115 (.0538) -.1412 (.0532) -.3267 (.0542) AR(4) -.0096 (.0569) -.1501 (.0537) -.3325 (.0542) -.0417 (.0572) .0124 (.0199) .1275 310 37.456 const. R T SSR 2 .0054 (.0212) .00003 313 43.854 .0072 (.0210) .0225 312 42.653 .0102 (.0199) .1243 311 37.860 2 (e) Using BIC (SBIC in Stata), find which is the best model in (c) and (d), respectively. TABLE 4: BIC Lag 1 2 3 4 U nemp .9024 .9166 .9148 .8238* U nemp .9085 .8993 .8058* .8170 The AR(4) is the best model for U nemp, and the AR(3) for U nempt (f ) Conduct a DF test for the AR(1) of U nempt and U nempt at the 5% level, respectively. Are the AR(1) of U nempt and U nempt stationary? [Hint: Using estimates and standard errors from Stata, do DF test manually.] In DF test, H0 : 1 = 1 (nonstationary) vs H1 : 1 < 1 (stationary). U nempt : t-stat.= (.9687 - 1)/.01604 = -1.951 < -1.645 (Critical Value at 5%). We reject H0 . Thus, the AR(1) model is stationary. U nempt : t-stat.= (.05639 - 1)/.05653 = -16.69 < -1.645 (Critical Value at 5%). We reject H0 . Thus, the AR(1) model is stationary. Note: This is essentially a left-tail t-test. (g) Conduct an ADF test (with intercept only) for the AR(3) of U nempt at the 5% level. Is the AR(3) of U nempt stationary? In ADF test, H0 : nonstationary vs H1 : stationary Augmented Dickey-Fuller test for unit root Number of obs = 310 ----------- Z(t) has t-distribution ----------Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value -----------------------------------------------------------------------------Z(t) -12.254 -2.339 -1.650 -1.284 -----------------------------------------------------------------------------p-value for Z(t) = 0.0000 Since the p-value is nearly 0%, which is less than 5%, we reject the null hypothesis. This suggests that the AR(3) model is stationary. Alternatively, since the test statistic, -12.254, is less than the critical value, -1.650, we reject H0 . Note that this is a left-tail test, but not t-test. (h) Using the AR(1) of U nempt , forecast (predict) U nempt of 2010m4. The actual value of U nempt of 2010m4 turned out to be 0.2 %. Find a forecast error (FE) and a root mean square forecast error (RMSFE) From the sample, we find Y2010m3 = -.2. ^ So, the forecast is: Y2010m4|2010m3 = .005447 + 3 .05939Y2010m3 = .005447 + (.05939)(-.2) = -.0064 [%] ^ The forecast error (FE) is Y2010m4|2010m3 - Y2010m3 = -.0064 - .2 = -.2064 [%] 2 = .2064 [%] RMSFE is (-.2064) Extra Credit: In (e), find BIC manually. U nemp AR(1): ln(SSR/T ) + (p + 1)(ln(T )/T ) = ln(43.642/314) + (1 + 1)(ln(314)/314) AR(2): ln(SSR/T ) + (p + 1)(ln(T )/T ) = ln(43.264/313) + (2 + 1)(ln(313)/313) AR(3): ln(SSR/T ) + (p + 1)(ln(T )/T ) = ln(42.363/312) + (3 + 1)(ln(312)/312) AR(4): ln(SSR/T ) + (p + 1)(ln(T )/T ) = ln(37.843/311) + (4 + 1)(ln(311)/311) U nemp AR(1): ln(SSR/T ) + (p + 1)(ln(T )/T ) = ln(43.854/313) + (1 + 1)(ln(313)/313) AR(2): ln(SSR/T ) + (p + 1)(ln(T )/T ) = ln(42.653/312) + (2 + 1)(ln(312)/312) AR(3): ln(SSR/T ) + (p + 1)(ln(T )/T ) = ln(37.860/311) + (3 + 1)(ln(311)/311) AR(4): ln(SSR/T ) + (p + 1)(ln(T )/T ) = ln(37.456/310) + (4 + 1)(ln(310)/310) The AR(4) is the best model for U nemp, and the AR(3) for U nempt Appendix: Stata Outputs = -1.937 = -1.924 = -1.923 = -2.014 = -1.929 = -1.935 = -2.032 = -2.021 . tsset time , m o n t h l y time variable : delta : t i m e , 1984 m 1 1 month t o 2010 m 3 . generate (1 m i s s i n g . . graph7 dunemp = D . unemp value generated ) time , xlab ylab s (.) c(l) unemp corrgram dunemp , lags (10) -1 0 1 -1 0 1 LAG AC PAC Q P r o b >Q [ Autocorrelation ] [ Partial Autocor ] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 0.0593 0.0594 1.1161 0.2908 | | 2 -0.1555 -0.1597 8.8109 0.0122 -| -| 3 -0.3307 -0.3267 43.708 0.0000 --| --| 4 -0.0340 -0.0417 44.077 0.0000 | | 5 0.2207 0.1463 59.723 0.0000 |- |- 6 0.1107 -0.0117 63.672 0.0000 | | 7 0.2164 0.2848 78.809 0.0000 |- |-- 8 -0.0444 0.1036 79.448 0.0000 | | 9 -0.3693 -0.3596 123.81 0.0000 --| --| 10 -0.1720 -0.0863 133.47 0.0000 -| | . regress unemp L . unemp N u m b e r of obs F( 1, 312) Prob > F R-s q u a r e d A d j R-s q u a r e d Root MSE = 314 = 3647.34 = 0.0000 = 0.9212 = 0.9209 = .374 Source | SS df MS - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Model | 510.18639 1 510.18639 Residual | 43.6422777 312 .139879095 - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Total | 553.828668 313 1.76942066 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - unemp | Coef . Std . Err . t P >| t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - unemp | L1 . | .9687331 .0160404 60.39 0.000 .937172 1.000294 4 | _ cons | .1877781 .0963853 1.95 0.052 - . 0018693 .3774254 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . regress u ne m p L (1 / 2 ) . un e m p N u m b e r of obs F( 2, 310) Prob > F R-s q u a r e d A d j R-s q u a r e d Root MSE = 313 = 1806.14 = 0.0000 = 0.9210 = 0.9205 = .37358 Source | SS df MS - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Model | 504.131065 2 252.065533 Residual | 43.2636386 310 .139560124 - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Total | 547.394704 312 1.7544702 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - unemp | Coef . Std . Err . t P >| t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - unemp | L1 . | 1.047084 .0565519 18.52 0.000 .93581 1.158358 L2 . | - . 0808716 .0571968 -1.41 0.158 - . 1934147 .0316715 | _ cons | .2031179 .0984006 2.06 0.040 .0095003 .3967355 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . regress u ne m p L (1 / 3 ) . un e m p N u m b e r of obs F( 3, 308) Prob > F R-s q u a r e d A d j R-s q u a r e d Root MSE = 312 = 1211.74 = 0.0000 = 0.9219 = 0.9211 = .37087 Source | SS df MS - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Model | 499.995374 3 166.665125 Residual | 42.3630612 308 .137542406 - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Total | 542.358435 311 1.7439178 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - unemp | Coef . Std . Err . t P >| t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - unemp | L1 . | 1.056944 .056385 18.75 0.000 .9459951 1.167892 L2 . | - . 2244753 .0814709 -2.76 0.006 - . 3847853 - . 0641653 L3 . | .1431798 .0570996 2.51 0.013 .0308251 .2555345 | _ cons | .1493853 .1001379 1.49 0.137 - . 0476556 .3464263 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . regress u ne m p L (1 / 4 ) . un e m p N u m b e r of obs F( 4, 306) Prob > F R-s q u a r e d A d j R-s q u a r e d Root MSE = 311 = 1013.69 = 0.0000 = 0.9298 = 0.9289 = .35167 Source | SS df MS - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Model | 501.451192 4 125.362798 Residual | 37.8427685 306 .123669178 - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Total | 539.293961 310 1.73965794 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - unemp | Coef . Std . Err . t P >| t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - unemp | L1 . | 1.009807 .0540319 18.69 0.000 .9034856 1.116128 L2 . | - . 1530716 .0782423 -1.96 0.051 - . 3070327 .0008895 L3 . | - . 1861212 .0787774 -2.36 0.019 - . 3411352 - . 0311072 L4 . | .3233378 .0549669 5.88 0.000 .2151768 .4314988 | _ cons | .0454608 .0967223 0.47 0.639 - . 1448641 .2357857 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . regress dunemp L . dunemp N u m b e r of obs F( 1, 311) Prob > F R-s q u a r e d A d j R-s q u a r e d Root MSE = = = = = = 313 1.10 0.2943 0.0035 0.0003 .37551 Source | SS df MS - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Model | .155637642 1 .155637642 Residual | 43.8540134 311 .141009689 - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Total | 44.009651 312 .141056574 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - dunemp | Coef . Std . Err . t P >| t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - dunemp | L1 . | .0593856 .056526 1.05 0.294 - . 0518362 .1706075 | _ cons | .0054472 .0212272 0.26 0.798 - . 0363198 .0472143 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . regress dunemp L (1 / 2 ) . d u n e m p N u m b e r of obs F( 2, 309) Prob > F R-s q u a r e d A d j R-s q u a r e d Root MSE = = = = = = 312 4.57 0.0110 0.0288 0.0225 .37153 Source | SS df MS - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Model | 1.26286489 2 .631432445 Residual | 42.6530034 309 .13803561 - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Total | 43.9158683 311 .14120858 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - dunemp | Coef . Std . Err . t P >| t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 5 | | .0661495 .0561277 1.18 0.239 - . 0442913 .1765904 | - . 1597355 .0560497 -2.85 0.005 - . 2700229 - . 0494481 | _ cons | .0072283 .0210391 0.34 0.731 - . 0341697 .0486262 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - dunemp L1 . L2 . . regress dunemp L (1 / 3 ) . d u n e m p N u m b e r of obs F( 3, 307) Prob > F R-s q u a r e d A d j R-s q u a r e d Root MSE = = = = = = 311 15.67 0.0000 0.1328 0.1243 .35117 Source | SS df MS - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Model | 5.79835069 3 1.93278356 Residual | 37.8599154 307 .1233222 - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Total | 43.6582661 310 .140833116 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - dunemp | Coef . Std . Err . t P >| t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - dunemp | L1 . | .0114654 .0537723 0.21 0.831 - . 0943436 .1172744 L2 . | - . 1412366 .0532115 -2.65 0.008 - . 2459419 - . 0365313 L3 . | - . 3266663 .0541591 -6.03 0.000 - . 4332363 - . 2200963 | _ cons | .0102198 .0199211 0.51 0.608 - . 0289794 .0494189 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . regress dunemp L (1 / 4 ) . d u n e m p N u m b e r of obs F( 4, 305) Prob > F R-s q u a r e d A d j R-s q u a r e d Root MSE = = = = = = 310 12.29 0.0000 0.1388 0.1275 .35044 Source | SS df MS - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Model | 6.03518011 4 1.50879503 Residual | 37.45579 305 .122805869 - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Total | 43.4909701 309 .140747476 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - dunemp | Coef . Std . Err . t P >| t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - dunemp | L1 . | - . 0096228 .0569536 -0.17 0.866 - . 1216946 .102449 L2 . | - . 1501377 .0536946 -2.80 0.005 - . 2557965 -.044479 L3 . | - . 3324914 .05419 -6.14 0.000 -.439125 - . 2258578 L4 . | - . 0416683 .0571849 -0.73 0.467 - . 1541952 .0708585 | _ cons | .0124501 .0199181 0.63 0.532 - . 0267441 .0516443 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . varsoc unemp , maxlag (4) S e l e c t i o n -o r d e r c r i t e r i a Sample : 1984 m 5 - 2010 m 3 N u m b e r of obs = 311 + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -+ | lag | LL LR df p FPE AIC HQIC SBIC | | - -+ - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - -- - - - - -- - - -- - - -| | 0 | -526.888 1.74525 3.39478 3.39958 3.4068 | | 1 | -134.665 784.45 1 0.000 .140999 .878872 .888485 .902922 | | 2 | -133.828 1.6741 1 0.196 .141147 .87992 .89434 .915996 | | 3 | -130.411 6.8333 1 0.009 .13897 .864379 .883605 .912479 | | 4 | -113.752 33.318 * 1 0.000 .125657 * .763678 * .787711 * .823804 * | + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -+ Endogenous : unemp Exogenous : _ cons . varsoc dunemp , maxlag (4) S e l e c t i o n -o r d e r c r i t e r i a Sample : 1984 m 6 - 2010 m 3 N u m b e r of obs = 310 + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -+ | lag | LL LR df p FPE AIC HQIC SBIC | | - -+ - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - -- - - - - -- - - -- - - -| | 0 | -135.448 .141202 .88031 .885128 .892363 | | 1 | -135.076 .74438 1 0.388 .141775 .88436 .893997 .908467 | | 2 | -130.637 8.8767 1 0.003 .138664 .862177 .876633 .898338 | | 3 | -112.562 36.151 * 1 0.000 .1242 * .752012 * .771286 * .800226 * | | 4 | -112.292 .53918 1 0.463 .124787 .756724 .780816 .816991 | + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -+ Endogenous : dunemp Exogenous : _ cons . dfuller dunemp , drift lags (3) test for unit root Number of obs = 310 Augmented D i c k e y -F u l l e r - - - - - - Z ( t ) h a s t-d i s t r i b u t i o n - - - - - - - - - - - - - - - - Test 1% C r i t i c a l 5% C r i t i c a l 10% C r i t i c a l Statistic Value Value Value - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Z(t) -12.254 -2.339 -1.650 -1.284 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - p-v a l u e f o r Z ( t ) = 0 . 0 0 0 0 . 6 ...
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