第5章　回歸程的函數(shù)形式

1、第5章回歸方程的函數(shù)形式the function forms of regression modelcontents log-linear model: measure elasticity (double log model) semi-log model log dependent variable: measure growth (log-lin model) log independent variables (lin-log model) hyperbolic model polynomial model summarylog-linear model:measure elastic

2、ity121212sometimes, well meet the model like the following:then how to estimate the model in ols?you see, in the model, dependent variable is notlinear with the parameters and .uyaxxeylog-linear model:measure elasticity 1122*01122*but, if we transfrom the model by log two sides simutaneously, we get

3、 a new model as following:lnlnlnlnnow, we denote ln,ln,ln,lnthen we can rewrite the model asyaxxuyyaxxxxy*01122which is just general multiple regression modelwhat we have learned in last chapter.xxulog-linear model:measure elasticity 01122 lnlnlnthe model above is often called in the log-linear mode

4、l, the partialcoefficients are the elasticity of over .for example, in the myxxuyxslog-linear model or double log model. 111111111odel above,ln lnwhich just denote the elasticity of over .that is, change 1%, will change %.dyxdy ydydxdxxdxyyxxylog-linear model: example in the end of 1920s, american m

5、athematician 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 = alak, the econometric model will be y = alak euwhere y denote the production, gdp, for example, l denote labor in

6、put to the production, and k denote total capital. a is the production elasticity of labor, is the production elasticity of capital.log-linear model: example y = alak 11we cant estimate the model directly, but wecan transform the model to log-linear model.lnlnlnlnwhere is the error term and is normd

7、ydldylalkl yyldlydydkdykal kk yykdkyyalkuuaaaaaal distribution.then we can use ols to estimate the new model.just the production elasticity of laborjust the production elasticity of capitalexample : the original datayearylkyearylk1955 11404393101821131965212323117463157151956 12041085291937491966226

8、977115213376421957 13470587382051921967241194115403635991958 12918789522151301968260881120663918471959 13996091712250211969277498122974223821960 15051195692370261970296530129554550491961 15789795272488971971306712133384846771962 16528696622606611972329030137385205531963 17849110334275466197335405715

9、9245615311964 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

10、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 estim

11、ated regression the 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=1719.46 remark:the production elasticity of labor is 0.3397, that is, if labor increase 1%,the production w

12、ill 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 datayeardemandgdppriceyeardemandgdpprice196054.154.1111.9197297.294.395.6196155.456.4112.4197310010010019

13、6258.559.4111.1197497.3101.4120.1196361.762.1110.2197593.5100.5131196463.665.9109197699.1105.3129.6196566.869.5108.31977100.9109.9137.7196670.373.2105.31978103.9114.4133.7196773.575.7105.41979106.9118.3144.5196878.379.9104.31980101.2119.6179196983.383.8101.7198198.1121.1189.4197088.986.297.7198295.6

14、120.6190.9197191.889.8100.3example 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 ene

15、rgy demand function. here, we use the log-linear model as following: ln(demand) = 0 1 ln(gdp) + 2 ln(price) + u1 is the income elasticity of demand.2 is the price elasticity of demandexample 5.3: the estimated regression the estimated model:ln(demand) =1.54950.9972 ln(gdp)0.3315ln(price) se =(0.0901

16、) (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=1693.67 remark: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,

17、 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 model there are two kinds of semi- lo

18、g model: the dependent variable is logged and the independent variables is logged. log dependent variable (log-lin model) ln(y) = 0 1 x1 + 2 x2 + u log independent variables (lin-log model)y = 0 1 ln(x1) + 2 lnx2) + ulog dependent variable:measure growth ratew see a simple regression modelln(y) = 0

19、1 x + uw what does the coefficient of x mean?we know that 1 is partial deviation of ln(y) to x, that is,so, the 1 means that independent variable x change 1 unit, the variable y will change 1001 %.note here the x is absolute value and y is the relative value. that is, 1 is the growth rate of y when

20、x change 1 unit. 1lndydy ydxdxexample: wage determinationw the estimated model:log(wage)=0.2840.092educ+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=80.39w remark:the partial coefficient for educ is 0.092, which mean

21、s 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 growthw let y0 stand for the value of gdp in 1960 and assume the average gdp growth rate is r, then the

22、 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. (*)w now, 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.

23、 what should we do?we logged equation (*) two sides simultaneously, and get, ln(yt)=ln(y0) + tln(1+r)example: 1960-1982 gdp growthw the corresponding econometric model isln(yt)=ln(y0) + tln(1+r) + ulet 0 = ln(y0), and 1= ln(1+r)rw the the model can rewrite asln(yt)= 0 + 1 t + unow, you see, the part

24、ial coefficient 1 stand for the growth of y when t change 1 unit. that is, 1 is just the growth rate of real gdp.the data for real gdp (yt)yeartgdpyeartgdpyeartgdp1960054.11968879.9197616105.31961156.41969983.8197717109.91962259.419701086.2197818114.41963362.119711189.8197919118.31964465.919721294.3

25、198020119.61965569.5197313100198121121.11966673.2197414101.4198222120.61967775.7197515100.56080100120gdp19601965197019751980yearthe scatter between real gdp and t6080100120gdp0510152025texample: 1960-1982 gdp growthw using the data for real gdp from example 5.3, we estimate the modelln(t)= 4.044 + 0

26、.0382 tse =(0.0158) (0.0012)t = (255.38) (30.97)n=23, r2=0.9786 adj-r2=0.9776 f=958.96w remark: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 c

27、alculate the compound growth rate by ln(1+r)=0.0382, therefore,r=e0.0382-1=1.0389-1=0.0389=3.89%.linear trend model6080100120gdp0510152025tw using 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=3352.72

28、w remark: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 modelsometimes, we meet the model with independent variables logged, such asy = 0 1 ln(x1)

29、+ 2 lnx2) + uthen what are the meaning of the partial coefficients?for example,so, 1 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.1111111, lndxdydydydxdxxx1/100.the relation between usa gnp (gnp

30、) and money supply (m2)gnp = 0 1 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 model t

31、he hyperbolic model is like the followingy = 0 1 1/x + u where 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, 0 is the asymptotic value or limited value of y.hyperbolic mod

32、el, cont.u = 0 1 1/xuwith different sign for the coefficient, the estimated equation will have different curve.xy001 00xy000average fixed cost functionengel expenditure curvephilips curvexy00, 1 0, so when unemployment rate increase, the income growth rate will decrease, which meet the theory.vtc= 0

33、 1 q + 2 q2 + 3 q3 + utcqcostv 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.vjust now, we use the hyperbolic function to model the philips curve, now we use a polynomial function to remodel the p

34、hilips curve. now, we specify the model as y=0 1 x + 2 x2 + uvuse the same data in table 5-6, we estimate the polynomial model above. =23.53-7.24 x + 0.34 x2 se=(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/x se=(1.0086) (4.6795) t = (-0.26) (4.40) n=12 r2=0.6594 adj-r2=0.6253