Lecture 2 Macroeconomic Model of China by Simultaneous Equations

Transcription

1 Lecture 2 Macroeconomic Model of China by Simultaneous Equations January 8, Introduction Simultaneous-equation Modeling the Chinese Macro-economy Aggregate Demand Model Determination of the price level P and inflation... 错误! 未定义书签 2.3 Aggregate Supply and the Production Function Combining aggregate demand with the production function Conclusions VEC Models... 错误! 未定义书签 3.1 VEC Model for lny, lnc and lni... 错误! 未定义书签 3.2 VEC Model for lny2, lnc, lni and lnpf omitting 错误! 未定义书签 3.3 VEC Model by adding M1 to the model above... 错误! 未定义书签 4. Forecasts for 错误! 未定义书签 5. Policy Analysis... 错误! 未定义书签 References

2 Lecture 1 Macroeconomic Model of China by Simultaneous Equations Gregory C. Chow Princeton University 1. Introduction This paper is a progress report on our initial effort to construct a macro econometric model of the Chinese economy. Our modeling effort follows the principles of from small to large, or from simple to complex and of basing econometric formation on previously confirmed economic hypotheses. The paper consists of three simple models: 1) a two equation model explaining consumption C real investment (accumulation) I both in real terms, the former by the permanent income hypothesis and the latter by the accelerations principle; 2) an equation to explain inflation by past inflation and past ratio of money supply to output, with an error-correction term capturing the deviation of the price level in the last period from its equilibrium level determined by the past ratio of money supply to output, and 3) a Cobb-Douglas production function determining output by capital, labor and total factor productivity TFP. We then combine the third model with the first by introducing the past deviation of output from capacity output determined by the production function as an error-correction term to explain output in addition to the predetermined variables in the first model of aggregate demand. We next generalize the above models to vector autoregression VAR models with cointegrating vectors, known as vector error-correction VEC models. Such models will be briefly explained before they are estimated. We will report on how well these models can explain recent data up to 2005, what forecasts for annual changes in output, consumption, investment and the price level they will produce from 2006 to 2010, and the effects of changes on monetary aggregates on these forecasts as an assessment of monetary policy. 2. Simultaneous-equation Models of the Chinese Macro-economy 2.1 Method of Simultaneous Equations y column vector of G endogenous variables; z column vector of K predetermined variables. Structural Equations By t = Γz t + u t Cov(u t ) = Σ Reduced-form Equations y t = B -1 Γz t + B -1 u t = Πz t + v t Cov(v t ) = BΣB for transpose All variables in y t are correlated with the residuals u t because from the reduced-form u t = Bv t and y t are correlated with v t. Hence we cannot obtain consistent estimate of the coefficients of the structural equations by the method of least squares consistency of 2

3 estimators by the method of least squares requires that all explanatory variables be uncorrected with the residuals. To eliminate the correlations of the dependent variables with the residuals u t we first regress all dependent variables by all predetermined variables, i.e., estimate the reduced form equations, and obtain the predicted values y* t of the endogenous variables from the estimated reduced form equations. In the second stage we apply least squares to estimate the structural equations after replacing y t by y* t. This method for estimating the structural equations is known as two-stage least squares TSLS. 2.2 Aggregate Demand Model This model was first estimated in Chow (1985) and further elaborated in Chow (2002, chapter 6). The consumption function is based on the permanent income hypothesis of Hall (1978), which states that real consumption C is a linear function of C(t-1) with a coefficient close to one and does not depend on any other variables observed in t-1. The reason is that, if consumption depends on permanent income, C(t-1) is the best measure of permanent income in period t-1. No other information available in t-1 can improve upon C(t-1) to serve as a measure of permanent income in t-1 and therefore for the prediction of C(t). If C(t) = C(t-1) + c + u(t) (1) where u(t) is a random residual, C(t) is said to follow a random walk with drift equal to the constant c. Chow (1985) confirmed this hypothesis using Chinese data from 1952 to Note that most of this period was under central planning. Chow (1985) discusses why the permanent income hypothesis applies to China during this period. A skeptical reader should examine the data carefully in order to be convinced. Our current effort has updated this model in log-linear form to 2005 and found that it explains the data fairly well as documented below. The statement that no other variables observed in t-1 than C(t-1) can help predict C(t) is true only approximately, as is known in the literature (see Chow and Zhang, 2002) and as shown in the estimated consumption functions reported below. The investment equation is based on the accelerations principle, which states the investment, being the rate of change in capital stock, is explained not by the level of income or output but by the rate of change in income. This theory is explained in detail in Chow (2002, chapter 6). Let desired capital stock K*(t) at the end of a year t be a linear function of income or output Y(t) in constant prices, or K* = a + by, and let the actual change in capital stock K(t) K(t-1) be a fraction β of (K*(t) - K(t-1)), implying K(t) = β [a+by(t)] + (1- β ) K(t-1). Since investment I is defined as K(t) (1-d) K(t-1), where d is the rate of depreciation for capital stock, subtracting (1-d)K(t-1) = (1-d)[ β(a+by(t-1) + (1- β) K(t-2)] from the above equation for K(t) yields the investment equation I(t) = cβa + βb[y(t)- dy(t-1)] + (1- β) I(t-1) (2) 3

4 An important implication of the accelerations principle to be tested is that if we explain investment I(t) by income Y(t) and lagged income Y(t-1), the coefficient of Y(t-1) should be negative and equal in absolute value to the coefficient of Y(t) if I denotes net investment (with d =0), or to (1-d) times the coefficient of Y(t) if I denotes gross investment, with d denoting the annual rate of depreciation. This equation is confirmed in Chow (1985). Our current study to update Chow (1985) to 2005 also confirms this hypothesis when the variables are in logs as described below. In the current model we choose to use lnc and lni as two dependent or endogenous variables. The consumption function assumes lnc to be a linear function of lnc(t-1) under the permanent income hypothesis, where Y(t) is defined as either Y1(t)=C(t) + I (t), or as Y2(t) = Y1(t) + [Ex(t) Im(t)]. We expect the coefficient of any lagged variable to be zero (or small) in explaining lnc(t) once the variable lnc(t-1) which represents permanent log income in the last period is included. If we apply the method of two-stage least squares to estimate the above consumption function, and denote by lny(t)* the estimate of logy(t) by lnc(t-1) and lni(t-1) we should expect the coefficient of lny(t)* (which can be observed in t-1) to be insignificant in explaining lnc(t) once lnc(t-1) is included as an explanatory variable, as we have actually found. The data are defined and constructed as follows. Up to 1982, the data are the same as in Chow (1985). Beginning in 1983, the data are linked with the official data provided in China Statistical Yearbook The consumption and investment series in Yearbook 2006 are linked to the Chow (1985) series by a factor of proportionality equal to the ratio of the Chow (1985) value to the Yearbook 2006 value in Y1 is the sum of C and I, while Y2 equals Y1 up to 1982 and includes (Export-Import) for years from 1983 onward. The price index up to 1982 is the retail price index used in Chow (1985). After 1983 it is the GDP deflator linked to the former series based on the ratio of the two series in Table 1 Data for Aggregate Demand Real GDP(1) Y=C+I Real GDP (2) Real Consumption Real Retail Price Index (1950=1) Investment Y1 Y2 RC RI RPI M1 M

5

6 Source: Statistical Yearbook of China, 2006; Chow(1985) 1 Estimation of Log-linear Model, Data from 1952 to 2005: We apply the method of two stage least squares 2SLS to estimate the model. Since the variable lny entering the consumption and investment functions may be correlated with the residuals of these equations, in the first stage of 2SLS we estimate lny by a regression on the predetermined variables lnc(t-1) and lni(t-1), denoting the result by lny*. In the second stage we estimate the consumption and investment equations with the variable lny in these equations replaced by lny*. The results of estimating the equations for lny, lnc and lny are given in Table 2, with standard errors in parentheses. Table 2 Estimation of the Aggregate Demand Model, lny1(t)* lnc(t) lni(t) lnc(t-1) (.0453) (.1550) lni(t-1) (.0376) (.0258) lny1*(t) (.1460) lny1*(t) lny1(t-1) (.8276) Intercept (.0860) (.0320) (.1282) R s DW lny2(t)* lnc(t) lni(t) lnc(t-1) (.0443) (.1696) lni(t-1) (.0367) (.0261) lny2*(t) (.1570) lny2*(t) lny2(t-1) (.7767) Intercept (.0840) (.0275).0085(.1265) R Consumption, Accumulation and Price data from 1952 to 1982 are from Chow(1985, 2002). The data from 1983 to 2005 are from China Statisticas Yearbook (2006), which are linked to the old data series by a factor of proportionality equal to the ratio of the two series in The series Real GDP(1) equals the sum of Consumption (RC) and Accumulation (RI) (Y1=RC+RI) The series Real GDP(2) from 1952 to 1982 equals Real GDP(1). From 1983 to 2005, these data are from China Statistical Yearbook 2006, which are linked to the old data series by a factor of proportionality equal to the ratio of the two series in The means that the data from 1983 on include net import. From 1952 to 1982, the price index is the Retail Price Index with 1950=100 as used in Chow (1985). During this period the consumer price index CPI, the retail price index RPI and the GDP deflator moved together. From 1983 on, our price index is the GDP deflator linked with the previous retail price index. 6

7 S DW In the equation to explain lny1(t) there is an error of about 7.4 percent, as compared with 3.8 percent in the equation for lnc and 23.8 percent in the equation for lni. In the equation to explain lny2(t) there is an error of about 7.3 percent, as compared with 3.8 percent for lnc and 24.0 percent for investment. The consumption function estimated by 2SLS confirms the permanent income hypothesis since the coefficient of lnc(t-1) is insignificantly different from 1 and the coefficient of logy* is insignificantly different from zero. In the model explaining lny1, the residuals of this equation for consumption are plotted in Figure 1, with the observed and estimated values of consumption given in Table 3. These residuals show that the equations fit the data well, and in particular the estimated values for the years are within one standard error of the regression , or about a 3.84 percent. Note that the large negative residual for 1990 might be due to the aftermath of the Tiananmen Incident. Figure 1 Prediction Errors of the Consumption Function LOG(RC) Residuals Table 3 Residuals of the Consumption Function obs Actual Fitted Residual Residual Plot * * * * * * * * * * * *. 7

8 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *. The estimated investment equation confirms the accelerations principle. When the variables lny* and lny(t-1) are introduced separately their coefficient estimates are respectively (1.6249) and (1.9628) when Y1 is used, and respectively (1.2081) and (1.4441) when Y2 is used. The point estimates of these coefficients have the right signs and relative magnitudes although the t ratios are small. As in Chow (1985) for the linear case, we use the difference [lny*-lny(t-1)] as one variable and find it to be highly significant and consisting with the accelerations principle. However the standard error of the regression is when Y1 is used and.2397 when Y2 is used, showing a large percentage error in the prediction of investment. For the model explaining lny1, Figure 2 is a graph of the prediction errors while Table 4 gives 8

9 their values numerically. The prediction errors are.343 in 1958,.456 in 1959, in 1961, in 1962,.367 in 1963, in 1967, and.385 in During this period covering the Great Leap followed by the Cultural Revolution, the equation fits very poorly, as does the production function to be discussed later. Hence there is good reason to omit these abnormal years for the purpose of estimating our models. Figure 2 Prediction Errors of the Investment Function LOG(RI) Residuals Table 4 Residuals of the Investment Equation Actual Fitted Residual Residual Plot * * * * * * * * * * * * * * * * * * * * * * * * * * 9

10 * * * * * * * * * * * * * * * * * * * * * * * * * * *. We have tried to examine the possible effects of M0 (currency in circulation), M1 and M2 (all divided by our price index) on consumption, investment and income by adding each of these variables separately in the equations of Table 2, but found none statistically significant. 2.3 Aggregate Supply and the Production Function This model updates the production function of Chow (2002, chapter 5) which is based on Chow and Lin (2002). Chow and Lin (2002) itself updated the model of Chow (1993). Both Chow (1993) and Chow and Lin (2002) or Chow (2002, chapter 5) found the elasticity of output respect to capital to be about 0.6 and the elasticity of output respect to labor about 0.4, no increase in total factor productivity before 1979, and an annual increase in TFP of about after To update this production function we use the same method as described in Chow (2002, chapter 5) to construct a series for capital stock up to To quote Chow (2002, pp. 92-3): After 1980 we adopt the following method to convert nominal into real gross capital formation in order to construct a capital stock series. First, the ratio of nominal to real GDP provides a GDP price deflator. This deflator is used to deflate the sum of nominal consumption and gross capital formation to obtain real domestic final expenditures in 1978 prices. Second, we convert nominal consumption to real consumption by using the 10

11 general consumer price index (CSY99, p.294), which is linked with the general retail price index (p. 294) for the period Third, we subtract real consumption from real domestic final expenditures to obtain real gross investment I (including inventory investment). We then construct our capital series K for the period based on the equation: K(t)=(1-d)[K(t-1)-720]+I(t)+720. The depreciation rate d equals 0.04, which is slightly lower than the average depreciation rate of non-land fixed capital found in China Report: Social and Economic Development , published by China Statistical Information and Consultancy Service Center, We use a slightly lower depreciation rate because our K includes inventory. We have thus obtained a capital stock series to update the production function of Chow (1993). The data on output Y, capital K and labor L are given in Table 5. Table 5 Production Function Data Year Y2 L K t

12 Source: Statistical Yearbook of China, 2006; Chow(2002) Note:Y1 and K were accounted in terms of the constant price of We present below an estimated Cobb-Douglas production for the period , omitting The abnormal years of the Great Leap and the Cultural Revolution were omitted from the sample because, in Chow (1993), a scatter diagram of ln(y/l) against ln(k/l) shows that the data fit a straight line well except for observations for these years. The variable t equals zero up to 1978, equals 1 in 1979 and increases by 1 each year afterwards. Chow (1993) found no increase in TFP up to The model allows for a constant rate of increase in TFP as measured by the coefficient of t. A low value of the Durban-Watson statistic is expected if we interpret this equation as a conintegration equation. ln(y2) = (.1232)*lnL (.0513)*lnK (.0025)*t R 2 =.9985 s=.0456 DW=.6345 (5) 12

13 Note the sum of the estimated coefficients of lnl and lnk to be almost exactly 1. Under the assumption that these coefficients sum to one, we re-estimate the production function by regressing ln(y/l) on ln(k/l) and T, obtaining ln(y2/l) = (.0987) (.0240)*ln(K/L) (.0022)*t R 2 =.9965 s=.0450 DW=.6361 (6) 2.4 Combining aggregate demand with the production function Let lnyd* be the estimate of lny in the aggregate demand model of section 2.2, namely, from a regression on lnc(t-1) and lni(t-1). Let lnpf be the estimate of lny by the production function, namely 0.6lnK + 0.4lnL t + constant. Define errorp as the difference lny lnpff, that is, the log of actual output minus the log of capacity output specified by the production. A positive value of errorp means that output is too high according to the amount specified by productivity. If errop(t-1) is high, it will assert a negative effect on current output through the error-correction effect similar to the one explaining inflation in section 2.2. Combining the error correction from the production side with factors affecting aggregate demand we have estimated an equation explaining lny from 1953 to 2005, omitting observations in , as follows ln(y2) = (.0733) (.0480)*lnC(-1) (.0416)*lnI(-1) (.1144)errorp(-1) R 2 =.9990 s= DW= (7) The coefficient of the error correction term from the production side has a t ratio of over 3 and is highly significant. The standard error in predicting lny2 is now about 3.5 percent as compared with 7.3 percent for the equation in Table 2 without combining the production effect, although the latter includes the abnormal years which tend to raise the standard error. Adding the error correction term from the production function to the consumption and investment equation respectively gives lnc = (.0295) (.2733)lnY2* (.2941) lnc(-1)) (.0929)errorp(-1) R 2 =.9993 s= DW= (8) lni = (.0364) (.5692)(lnY2*-lnY2(-1)) (.0132)lnI (.2863)errorp(-1) R 2 =.9946 s = DW = (9) 13

14 In the consumption function (8) the error correction term from the production function is almost zero, confirming the permanent income hypothesis, whereas the term is highly significant in the investment function (9). 2.5 Conclusions 1. We have estimated a model to explain log consumption by the permanent hypothesis and log investment by the accelerations principle. The model explains recent data on consumption and investment. 2. We have also updated the Chow (1993) and Chow-Lin (2002) aggregate production function using data up to We have combined the aggregate demand model of 1 and the aggregate supply model of 3 to explain output by aggregate demand supplemented by productive capacity through an error-correction mechanism. References Chow, Gregory C. (1985) A model of national income determination in China, Journal of Political Economy. (1987), Money and price level determination in China, Journal of Comparative Economics. (1993)Capital formation and economic growth in China, Quarterly Journal of Economics., China s Economic Transformation. Oxford: Blackwell, and Anloh Lin (2002), Aggregate production functions for Taiwan and Mainland China: a comparative study, Journal of Comparative Economics and Yan Shen (2006), Money, output and price level in the Chinese macro-economy, Asian-Pacific Journal of Accounting and Economics. and Zhang (2002), Equity premium and consumption sensitivity in the context of robust control, Journal of Economic Dynamics and Control. Hall, Robert (1978), A permanent income hypothesis of consumption, Journal of Political Economy. 14