第5章-回归方程的函数形式课件

第5章回归方程的函数形式The function forms of regression modelContentsLog-linear model:measure elasticity(double log model)Semi-log modelLog dependent variable:measure growth(log-lin model)Log independent variables(lin-log model)Hyperbolic modelPolynomial modelsummaryLog-linear model:Measure elasticityLog-linear model:Measure elasticityLog-linear model:Measure elasticityLog-linear model:exampleIn the end of 1920s,American mathematician Charles Cobb and economist Paul Dauglas put forward the famous production function,that is,Cobb-Dauglas production function.The C-D production function is as followingY=ALaKb,the econometric model will be Y=ALaKb euWhere Y denote the production,GDP,for example,L denote labor input to the production,and K denote total capital.a is the production elasticity of labor,b is the production elasticity of capital.Log-linear model:exampleY=ALaKbJust the production elasticity of laborJust the production elasticity of capitalExample:the original datayearYLKyearYLK1955 11404393101821131965212323117463157151956 12041085291937491966226977115213376421957 13470587382051921967241194115403635991958 12918789522151301968260881120663918471959 13996091712250211969277498122974223821960 15051195692370261970296530129554550491961 15789795272488971971306712133384846771962 16528696622606611972329030137385205531963 178491103342754661973354057159245615311964 19945710981295387197437497714154609825Example(textbook,ex5.2,p105):the transformed datayearln(Y)ln(L)ln(K)yearln(Y)ln(L)ln(K)195511.64 9.14 12.11 196512.27 9.37 12.66 195611.70 9.05 12.17 196612.33 9.35 12.73 195711.81 9.08 12.23 196712.39 9.35 12.80 195811.77 9.10 12.28 196812.47 9.40 12.88 195911.85 9.12 12.32 196912.53 9.42 12.95 196011.92 9.17 12.38 197012.60 9.47 13.03 196111.97 9.16 12.42 197112.63 9.50 13.09 196212.02 9.18 12.47 197212.70 9.53 13.16 196312.09 9.24 12.53 197312.78 9.68 13.24 196412.20 9.30 12.60 197412.83 9.56 13.32 Example(textbook,ex5.2,):the estimated regressionThe estimated model:ln()=-1.652+0.3397 ln(L)+0.846 ln(K)se=(0.606)(0.186)(0.093)t =(-2.73)(1.83)(9.06)p=(0.014)(0.085)(0.000)n=20,R2=0.9951 Adj-R2=0.9945 F=1719.46Remark:The production elasticity of labor is 0.3397,that is,if labor increase 1%,the production will increase 0.3397%.In the same way,the production elasticity of capital is 0.846,thats,if capital increase 1%,the production will increse 0.846%.Example 5.3 Demand for energy:The original datayeardemandgdppriceyeardemandgdpprice196054.154.1111.9197297.294.395.6196155.456.4112.41973100100100196258.559.4111.1197497.3101.4120.1196361.762.1110.2197593.5100.5131196463.665.9109197699.1105.3129.6196566.869.5108.31977100.9109.9137.7196670.373.2105.31978103.9114.4133.7196773.575.7105.41979106.9118.3144.5196878.379.9104.31980101.2119.6179196983.383.8101.7198198.1121.1189.4197088.986.297.7198295.6120.6190.9197191.889.8100.3Example 5.3 Demand for energy(textbook,p106)Example 5.3 give the data about the energy demand of seven OECD countries during 19601982.In the table,demand is the total demand for energy,gdp for real GDP,and price for real energy price.Now,we want to estimate the energy demand function.Here,we use the log-linear model as following:ln(demand)=b0+b1 ln(gdp)+b2 ln(price)+ub1 is the income elasticity of demand.b2 is the price elasticity of demandExample 5.3:the estimated regressionThe estimated model:ln(demand)=1.5495+0.9972 ln(gdp)0.3315ln(price)se =(0.0901)(0.0191)(0.0243)t =(17.20)(52.17)(-13.63)p =(0.000)(0.000)(0.000)n=23 R2=0.994 Adj-R2=0.9935 F=1693.67Remark:The energy demand is positive related with real GDP and negative related with energy price,which accord with the economic theory.The demand elasticity of income is 0.9972,that is,when other factors fixed,if real GDP increase 1%,the demand for energy will increase 0.9972%.The demand elasticity of price is-0.3315,which means when income is fixed,if the price for energy increase 1%,the demand for energy will decrease 0.3315%.Semi-log modelThere are two kinds of semi-log model:the dependent variable is logged and the independent variables is logged.log dependent variable(log-lin model)ln(Y)=b0+b1 X1+b2 X2+ulog independent variables(lin-log model)Y=b0+b1 ln(X1)+b2 ln(X2)+uLog dependent variable:measure growth ratewSee a simple regression modelln(Y)=b0+b1 X+uwWhat does the coefficient of X mean?We know that b1 is partial deviation of ln(Y)to X,that is,So,the b1 means that independent variable X change 1 unit,the variable Y will change 100b1%.Note here the X is absolute value and Y is the relative value.That is,b1 is the growth rate of Y when X change 1 unit.Example:wage determinationwThe estimated model:log(wage)=0.284+0.092educ+0.0041exper+0.022tenure se =(0.1042)(0.0920)(0.0041)(0.0031)t =(2.73)(12.56)(2.39)(7.13)n=526 R2=0.3160 Adj-R2=0.3121 F=80.39wRemark:The partial coefficient for educ is 0.092,which means when the education increase 1 year with exper and tenure fixed,then the wage per hour will increase The same meaning to the partial coefficients for exper and tenure.9.2%.Example:1960-1982 GDP growthwLet Y0 stand for the value of GDP in 1960 and assume the average GDP growth rate is r,then the GDP in 1961 will beY1=Y0(1+r),the GDP in 1962 will beY2=Y1(1+r)=Y0(1+r)2,in the same way,the GDP for t year beyond the initial year will beYt=Y0(1+r)t.(*)wNow,we have the data for GDP(Y)during 19601982.(see example 5.3),we want to estimate the average growth rate of GDP during the period.What should we do?We logged equation(*)two sides simultaneously,and get,ln(Yt)=ln(Y0)+tln(1+r)Example:1960-1982 GDP growthwThe corresponding econometric model isln(Yt)=ln(Y0)+tln(1+r)+ulet b0=ln(Y0),and b1=ln(1+r)rwThe the model can rewrite asln(Yt)=b0+b1 t+uNow,you see,the partial coefficient b1 stand for the growth of Y when t change 1 unit.That is,b1 is just the growth rate of real GDP.The data for real GDP(Yt)yeartgdpyeartgdpyeartgdp1960054.11968879.9197616105.31961156.41969983.8197717109.91962259.419701086.2197818114.41963362.119711189.8197919118.31964465.919721294.3198020119.61965569.5197313100198121121.11966673.2197414101.4198222120.61967775.7197515100.5The scatter between real GDP and tExample:1960-1982 GDP growthwUsing the data for real GDP from example 5.3,we estimate the modelln(t)=4.044+0.0382 tse =(0.0158)(0.0012)t =(255.38)(30.97)n=23,R2=0.9786 Adj-R2=0.9776 F=958.96wRemark:The slope of the estimate equation is 0.0382,which means that the growth rate for real GDP is 3.82%every year in average.The estimated slope is often called instantaneous growth rate.But we can easily calculate the compound growth rate by ln(1+r)=0.0382,therefore,r=e0.0382-1=1.0389-1=0.0389=3.89%.Linear trend modelwUsing the data for real GDP from example 5.3,we estimate the modelt=50.3+3.277 tse=(0.776)(0.0566)t =(64.82)(57.90)n=23 R2=0.9938 Adj-R2=0.9935 F=3352.72wRemark:The slope means that t increase 1 unit,the real GDP will inrease 3.277,which is absolute value.That is,real GPD in this year will 3.277 greater than that of the year before this year.Linear trend modelLog independent variablesSometimes,we meet the model with independent variables logged,such asY=b0+b1 ln(X1)+b2 ln(X2)+uThen what are the meaning of the partial coefficients?For example,So,b1 means that when independent variable X1 changes 1%,the dependent variable will changeNote,the change of X1 is relative value and Y is absolute value.b1/100.ExampleThe relation between USA GNP(gnp)and Money supply(m2)gnp=b0+b1 ln(m2)+uwe estimate the modelgnp=-16329.21+2584.785 ln(m2)Se =(696.60)(94.0414)t =(-23.44)(27.49)n=15 R2=0.9831 Adj-R2=0.9818 F=755.46Remark:The slope means when money supply m2 increase 1%,the USA GNP gnp will increase 2584.785/100=25.85Hyperbolic modelThe hyperbolic model is like the followingY=b0+b1 1/X+uWhere X is nonlinear with Y,but Y is still linear with the parameters,so the OLS still work.But instead,we will take 1/X as a new independent variable.In the hyperbolic model,b0 is the asymptotic value or limited value of Y.Hyperbolic model,cont.u=b0+b1 1/XuWith different sign for the coefficient,the estimated equation will have different curve.XYb00b1 00XYb000Average fixed cost functionEngel expenditure curvePhilips curveXYb00,b1 0,so when unemployment rate increase,the income growth rate will decrease,which meet the theory.vTC=b0+b1 Q+b2 Q2+b3 Q3+uPolynomial modelTCQcostv In order to best fit the data,polynomial model may be a good choice.v For example,we can use model the total cost function as polynomial regression.Example:Philips curvevJust now,we use the hyperbolic function to model the Philips curve,now we use a polynomial function to remodel the Philips curve.Now,we specify the model asY=b0+b1 X+b2 X2+uCompare the two kinds of models vUse the same data in table 5-6,we estimate the polynomial model above.=23.53-7.24 X+0.34 X2se=(3.76)(1.54)(0.15)t=(6.26)(-4.71)(4.22)N=12 R2=0.8370 Adj-R2=0.8228 F=23.11vUsing hyperbolic model,estimation of the American Philips Curve is=-0.2594+20.5879 1/Xse=(1.0086)(4.6795)t =(-0.26)(4.40)n=12 R2=0.6594 Adj-R2=0.6253 F=19.36Example:Philips curvevLet see the scatter and fitted curve using polynomial model:Regression through the originvBecause there is no intercept,the OLS estimate will beRegression through the originvSeveral important characteristicsd.f.of sigma hat squred is n-k(not n-k-1),R-squared formula we mentioned before is not eligible in this case.However,we can still use squared the correlation coefficient of Y and its fitted value(Y hat).The sum of squared residuals is not zero.关于度量比例和单位vUsually,unit is not a problem in estimation.vHowever,because of computer calculation system,the scale of independent variables and dependent variable should be similar.标准化变量的回归vThe meaning of OLS estimate is how much standard deviation of Y changes when X change one standard deviation.summaryLog-linear model:measure elasticityln(Y)=b0+b1 ln(X1)+b2ln(X2)+u b1%Semi-log modelLog dependent variable:measure growthln(Y)=b0+b1 X1+b2X2+u 100 b1%Log independent variablesY=b0+b1 ln(X1)+b2ln(X2)+u b1/100Hyperbolic modelY=b0+b1 1/X1+uPolynomial modelY=b0+b1 X+b2X2+u