2. Unemployment ÍÍSs) „Auf die Arbeit schimpft man nur so lange, bis man keine mehr hat." Harry Sinclair Lewis (1856-1950) 26.10.09 phd course "Econometrics", Brno, ©W.G. Müller, JKU Linz Chapter 2 - Contents 2. Unemployment (Okun's Law - simple lineare regression inferential) 2.1 The Okun equation in first differences 2.2 Residual plots (assumptions on distributions and their test) 2.3 The coefficient of determination 2.4 ŕ-Test and p-values 2.5 Studentized residuals and Cook's distances 2.6 Reverse regression 2.7 Homework (Okun's Law in alternative forms) 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 2 (m. An eternal hot topic PSYCHOLOGISCH I! MONOGRAPH I IV N DIE ARBEITSLOSEN VON MARIENTHAL Jijnf fidJírOKEAPIIIŠ/ľliĽE VÜKSir™ 15HEII DIE WIBKUNC.R» LA ĹÍLlDAUERÍDliS ..IHHKľľfiT.OfiTCTKĽLT KIT ÍIÍĽK AMEAMG ZUH GE&CHItHTK nklrt SOÍIO(mAr]lIĽ BĽŕ.Ä^TTITÍ.T Iran HTtft^TľíUíEu^JAĽU VOJÍ ľJl7R USTĽEEĽÍJilSUHlíN WlK'ľStHAPtSPSYCIKJLOGISťimH POESCHV^tSírRIJJ! VERLAfr VO K S. Ulli/.Víl. IN LEIPÍJKj L í J j 26.10.09 phd course "Econometrics", Brno, ©W.G. Müller, JKU Linz 3 Okun's Law: (m. Negative but underproportional relationship between economic growth and increase of the rate of unemployment! Okun, A.M., "Potential GNP: Its measurement and significance", Co wies Foundation Paper 190, 1962. Arthur M. Okun (1928-1980) 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 4 Okun's equation in first differences: (has) ut- utl = 0.30 - 0.30g . For the reduction of the unemployment rate of 1 percentage point we require a growth of the economy of 1% /0.3 = 3.3% (the Okun coefficient). 26.10.09 Constant i ne ManosUSA Unemployed, 1957 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz The Potential Output (m. Level of output on which all resources are fully exhausted, here pt =YJ1 + 0.033(ur 4)]. Mchad Lenson Full Product i on and Full Errpl oyment under Cur Denver at i c Syst em of Private Enterprise, ca. 1944 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz Okun's regression Data: USA from second quarter 1947 until fourth quarter 1960 Unemployment Equation Specification Equation Specification: Dependent variable followed by list of regressors including AR MA and PDL terms. OR an explicit equation like Y=c(1 )+c(2)"X. Timet «ne* /X/Stai« Untj»ll<7>itjit Rale <&97) I I 0-5 Mi oi La* or I er« e B 5-1» Mi II - U 44 — is 2(■■/„ m n - is v« Method: | LS - Least Squares (N LS and AR MA) Sample: 3 1947:11960:4 1 ^ OK (J Options 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 7 2.2 Residual Plots ■' i''' i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■' 1948 1950 1952 1954 1956 1958 1960 -------Residual -------Actual-------Fitted 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 8 (m. Residual Scatterplots (m. 1 c, 1 ^ 1.0-0.5-LU Ľ 0.0--0.5--1 0- ■ L- L ■ 1.0-0.5-Q ifi LU Ľ 0.0--0.5-■1 n L ' ' ' ' L ■ ' L i r = - J r " e 1 1 1 1 [ ' ■"-■ 1--------------------------------1--------------------------------1--------------------------------1 4-2 0246 -1 0 1 2 GDPRP D(UR) against regressor and regressand. 26.10.09 phd course "Econometrics", Brno, ©W.G. Müller, JKU Linz Distribution Plots (m. Kent I DEi-slkCEparEEhrlfcH, h- aJEffl) 1.2 Kernel density and Q-Q Plot 26.10.09 phd course "Econometrics", Brno, ©W.G. Müller, JKU Linz 10 2.3 The Coefficient of Determination (p21ff) (m. is motivated from the decomposition of sum of squares in the linear regression: SST = SSR + SSE IJyt--y r = U?t--y i2 +LV. y V-ii my = ý V-u m ý+~e-e The fraction of the explained from the total variation is called the coefficient of determination or R2=l- i ~2 e: —\2 Z^r-37) = 1- e e y\I-iťlT)y 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 11 R2 in the output: T —k—í Dependent Variable: D(UR) Method: Least Squares Sample(adjusted): 1947:2 1960:4 Included observations: 55 after adjusting Variable Coefficient C GDPRP 0.311155 0.313616 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Statistic 058945 036429 5.27^767 -8.6090*5 0.0000 0.0000 ^262 -22.53384 1.646644 Mean dependent var S.D.dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 0043345 0.569696 0.892140 0.965134 74.11617 0.000000 26.10.09 phd course "Econometrics", Brno, ©W.G. Müller, JKU Linz 12 Correspondence with the Correlation Coefficient since A = xc*. ~*y then holds —\ 2 R2 = _I(ä-y)2_[IU-*)U-y)] _\2 _\2 ZU-?) ZU--*) Ti(yi-y) (m. In the simple linear regression holds: R2 = Korr(x,y)2. Since ß0 = y-ßxx we have yf-y = Á(*«-*) and Sfo-502 =#!(*,-*)2 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 13 Correspondence with the Correlation Coefficient (p23) (has) Always holds (in linear eegression) R2 = Korr(y,y)2. Define 0 M=I-ťl/T ~'7i,r0 ~'7i,r0 then we have y M y = y M y and thus 2 y M y yM y y M y RA = o o ^'li jtO y'M"y y'M"y y'M"y 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 14 2.4 ŕ -test and p -values (p23ff) Íífas) Basic Assumption/Fiction: one considers the existing data (pairs) as a random sample from a universe of possible configurations, that are generated from an economy in a constant state. —ME— SCIENTIFIC AMERICA Infinite Earths in Et«V VERS Really Exist Orphan Drugs: Too Successful? Keys to Robust Networks ■ illpox Defense 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 15 The Distribution of Estimators (p17) iiSs) E[ß IX] = EffX'X^X')/ IX] = EtfX'X^XYXytf+e,) IX] = E[ß +(XX)-]X's IX] = yß +(XX)-]X'E[s \X\=ß Var[/? IX] = Var[ß +(XX)-'Xe IX] = fX'Xj-^X'Varte \X]X(XX)' = a£2 (XXyXX(XX)1 = as2 (XX)1. ß = ß + (X'XylX'e: N(ß,(T2£(X'Xyl) 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 16 The Gau ß-M ar kov Theorem /Ä\ (1821,1912) p18 ílFASl Amongst all linear, unbiased estimators is the OLS-estimator the one with the smallest variance! fiW (Johann) Carl Friedrich Gauß (1777-1855) —, ÄHdpeü ÄHdpeeeuv MapKoe (1856-1922) 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 17 Derivation: Be ß* = Cy an unbiased estimator, then: 'E[/0*1 = E[CXß+Cs]} = CXß= ß, which yields: CX=I; so one gets Var[/?*] = Var[CXß +Ce] = CďlC = CC'ď Be C = (XX)-]X'+D; since CX=I we have DX=0; one yields CC' = (XTcy'+DD' and CC'-(ZTj;>0 Thus /? is of minimal variance. 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 18 (m. Statistical Tests íiSs) Are used for checking hypotheses, e.g. of the form H0: ßt = 0. hypothesis 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 19 What has that to do with Econometrics? (m. ß /"o^iX'xy^t ~ t-distribution, where V / = £ř ^ t2/(T-k-l) 26.10.09 phd course "Econometrics", Brno, ©W.G. Müller, JKU Linz 20 c GDPRP R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.583059 Mean dependent var 0.575192 S.D. dependent var 0.371312 Akaike info criterion 7.307262 Schwarz criterion -22.53384 F-statistic 1.646644 Prob(F-statistic) 0.043345 0.569696 0.892140 0.965134 74.11617 0.000000 26.10.09 phd course "Econometrics", Brno, ©W.G. Müller, JKU Linz 21 1 zrrors in Testing (p31f) Decision H0 accept H0 reject Status of the world H0 correct decision error of 1st kind (a) u„ error of 2nd kind iß) correct decision www .stauff .de/matgesch/dateien/feh erersterzweiterart.htm 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 2.5 Studentized Residuals and ... Under the given assumptions residuals are normally distributed with expectation 0 and variance <7e2(l-HJ. The matrix H=X(XX)1X' is called hat matrix, since it transforms the original values into the predicted values ý = Hy 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 23 (m. Vectormanipulation in Eviews: external and internal studentizing mtos(htt, hhtt) series rresid=resid/@sqrt(1 -hhtt)/okun.@se K = ' ^V1^ series sresid=rresid*@sqrt((okun.@regobs-2)/ (okun.@regobs-1 -rresidA2)) (m. stom(groupx,x) vector htt=@getmaindiagonal(x*(g)inverse((a)inner(x))* (g)transpose(x)) st=rt T-k-\ T-k-r2 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 24 externally studentized residuals (t-distributed) for Okun regression (m. 111111111111111111111111111111111111111111111111111111 47 48 49 50 51 52 53 54 55 56 57 58 59 60 SRESIĽ 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 25 Multiple Testing (m. Based on the Bonferroni inequality: P(AnB) > P(A) + P(B) - 1 Carlo Emilio Bonferroni (1892-1960) With many! tests benchmark 3.5, see Tlrj in> ■ i tfi 6rtm iTJ r j ■ • : TI i" ■■'■vr[£Y APPLIED LINEAR REGRESSION Sanford Weisberg 26.10.09 phd course "Econometrics", Brno, ©W.G. Müller, JKU Linz 26 Cook's distances (m. Distances in the parameter space, that arise when single observations get deleted: í TT \ 1 D,=---rt ' k + 1 2 H n \ \-H tt J i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | i i 1948 1950 1952 1954 1956 1958 1960 Dl R. Dennis Cook 26.10.09 phd course "Econometrics", Brno, ©W.G. Müller, JKU Linz 27 „Robustified" Estimation Dependent Variable: D(UR) Method: Least Squares Sample: 1947:2 1953:4 1954:2 1960:4 Included observations: 54 Variable Coefficient Std. Error t-Statistic Prob. C 0.277743 0.055130 5.037979 0.0000 GDPRP -0.298737 0.033789 -8.841121 0.0000 R-squared 0.600508 Mean dependent var 0.015130 Adjusted R-squared 0.592826 S.D. dependent var 0.534849 S.E. of regression 0.341288 Akaike info criterion 0.724154 Sum squared resid 6.056835 Schwarz criterion 0.797821 Log likelihood -17.55217 F-statistic 78.16543 Durbin-Watson stat 1.680899 Prob(F-statistic) 0.000000 ut- ut_j = 0.278- 0.299 gyt = -0.30 (gyt- 0.93). 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 28 (m. 2.6 Reverse Regression íiSs) Exchange regressor and regressand! Instead oiy=ß0+ß1xt+st use x = Ä0+Ájyt+^r The estimator is analogously given as X = (Y'Y) lY'x with Y={i, yj and thus l\ = R2/ßr Generalization: error-in-variables models \N\thgapt = yp/yt-l. The second suggested equation is log (100-uJ = ß0 + ßj log yt + fat. Generate Series by Equation -Enter equation- pgdp=471.025*1.035"((@trend-33.5)/4J - Sample - 1947:1 1960:4 OK Cancel 26.10.09 phd course "Econometrics", Brno, © W.G. Müller, JKU Linz 33