Lecture 3 Macroeconomic models by VAR

Transcription

1 Lecture 3 Macroeconomic models by VAR China s Econ Transformation, chapters 6 and 7. Lectures 2 and 3 as Word Documents. Chow, J of Comparative Economics Chow and Shen, Money, Price Level and output in the Chinese macro-economy, Asia Pacific J of Accounting and Economics, December, 2005.

2 What determine the price level P? Simultaneous equations of last lecture determine Y, C and I in real terms. Quantity Equation Mv = PY, v is velocity Quantity Theory of Money. P = v(m/y). If v changes slowly M/Y determines P. Exam question: What is the most important variable determining the price level P?

3 What determine the rate of inflation logp t logp t-1 = ΔlogP t? Δ log(m/y) t, Δ logp t-1 Estimate a regression logp t = a + b log(m/y) t + u t The deviation u t-1 of logp t-1 from its equilibrium level given by a + b log(m/y) t-1 has a negative effect on ΔlogP t. See Engle and Granger (1987) for the econometric theory for such models.

4 Outline This lecture presents an abridged history of China s inflation from the early 1950s to It explains inflation by a single equation with an error-correction mechanism deviation of actual from the equilibrium relation between log Price and Log Y and Log Money Supply. The equilibrium relation between logp, logy and logm is a cointegration relation of a VAR model. The basic hypothesis concerning the co-movements of these three variables is due to the work of Milton Friedman (1994). A major proposition guiding our work is that when money supply increases, whatever the cause, real output will first increase before the price level increases but real output will die down more rapidly than prices. This proposition will be used to explain the changes in the price level and output in relation to the changes in money stock in China s macroeconomic history from the 1950s to 2004.

5 II. A Brief Monetary History of the Chinese Macro-economy In this section we provide a brief explanation of the important movements of price and output in response to changes in money supply where money supply itself can be the result of other factors. This is a simplified economic history but it can be interesting to the extent that money supply changes can be explained and the resulting movements in price and output are consistent with Friedman s proposition as stated at the beginning of this paper. We start with an explanation of the large price changes, and then explain the large output changes.table 1 provides five sets of data from 1954 to 2002: General retail price index P at the end of the year (column 2), the inflation rate measured by 100 times P(t)/P(t-1) (column 3), real GDP index Y (column 4), M2 at the end of the year in 100 million yuan (column 5) and M1 at the end of the year (column 6).

6 Data on Inflation and its Determinants General Retail Price Index Price Index Preceding Year=100 GDP Index 1978=100 Currency in Circulation (100 million) End of Year

7

8 , , , , , , , , , , , , , , , , , , ,468.30

9

10

11

12 Error-correction model In the dynamic equation, y t is affected not only by x t but possibly by past values of both variables. In the long-run, let us assume that there is an equilibrium relation between the two variables given by y-α 0 -α 1 x=0. The dynamic equation has an error correction mechanism built into it. The correction mechanism specifies that a positive deviation u t-1 =y t-1 -α 0 -α 1 x t-1 from equilibrium in the last period will assert a negative effect on the change Δy t =y t -y t-1 of the dependent variable in the current period. Thus the coefficient γ of this deviation u t-1 in the regression of Δy t is negative. The dynamic equation proposed attempts to explain Δy t by Δx t, past changes of both x and y and by this error-correction term, i.e., Δy t =β 0 +β 1 Δx t +β 2 Δx t-1 +β 3 Δy t-1 +γu t-1 +ε t (1)

13 Equilibrium relation between ln(p) and ln(m2/y)

14 Estimated equilibrium equation If we fit a linear regression using the data of Figure 7.1 we obtain = + Adjusted R 2 = log( P) log( M / Y), (0.031) (0.0102) log( P) log( M ) log( Y) 2 (0.109) (0.033) (0.0729) 2 = + Adjusted 2 R =

15 Estimated error-correction equation to explain inflation To explain the inflation rate ΔlogP we use its own lagged value, the current and lagged values of ΔlogY (Y being real GDP) and ΔlogM2, as well as the lagged value of the residual of the cointegration equation u t-1 : Δ log( P) = Δlog M 0.182ΔlogY 2 (0.009) (0.058) (0.06) Δlog( P ) 0.036Δ log( M, ) Δlog( Y ) 0.2 u (0.108) (0.58) (0.07) (0.058), Adjusted The above is known as an error-correction equation because of the last term. As Δ log( M ) is not significant, it is omitted in the following regression: R = 0.661

16 Error correction equation first estimated in 1985 Chow (1987) estimated an error correction equation to explain the inflation rate Δln p in China for the period 1952 to 1985 using currency in circulation to measure M, as reported in the first edition of Chow (2007). Δlnp = Δln(M/Y) Δlnp u -1 R 2 = (2) ( ) (0.0201) (0.1098) (0.1209) Note that Δln(M/Y) -1 was omitted because its coefficient was found to be statistically insignificant. The deviation u from the long-run relation was the residual of a regression of lnp on ln(m/y) which was lnp = ln(m/y)

17 VAR to explain three variables, log M, log P and log Y We will estimate a VAR to explain the changes in the three variables, log M, log P and log Y, denoted by the vector x. The VAR is a vector regression of Δx(t) on x(t-1) and Δx(t-1). By the maximum likelihood method of Johansen (1991), the coefficient matrix of x(t-1) is found to have rank one, to be written as ab. The vector b x(t-1) corresponds to the regression of log P on log(m/y) in section III, and turns out to be similar numerically. Using this VAR we compute the impulse responses of log P and log Y to unexpected changes in M2. The dynamic effects are found to be consistent with the major propositions of Milton Friedman stated above on the effects of money supply on price and output, and as recently summarized in Bernanke (2003). We compare the impulse responses estimated by using US and Chinese data with M1 replacing M2, and finds that the general patterns are quite similar in spite of the institutional differences between these two countries.

18 Vector Autoregression VAR A VAR for these three variables (considered as a vector, hence the name) may be the following: p t = a 0 + a 1 p t-1 + a 2 y t-1 + a 3 m t-1 + a 4 p t-2 + a 5 y t-2 + a 6 m t-2 + u 1t y t = b 0 + b 1 p t-1 + b 2 y t-1 + b 3 m t-1 + b 4 p t-2 + b 5 y t-2 + b 6 m t-2 + u 2t m t = c 0 + c 1 p t-1 + c 2 y t-1 + c 3 m t-1 + c 4 p t-2 + c 5 y t-2 + c 6 m t-2 + u 3t The main difference between a VAR and a simultaneous equation system is that it does not distinguish between endogenous variables and predetermined variables. Each variable is treated as an endogenous variable and explained by its own lagged values and the lagged values of all other variables in the system.

19 Use of VAR to test the Friedman proposition Denote by Δx the vector composed of Δlog (P), Δlog (M2) and Δlog (Y). Write a second-order VAR for x as x(t) = A 1 x(t-1) + A 2 x(t-2). To express this equation in first differences subtract x(t-1) from both sides. x(t) x(t-1) = A 1 x(t-1)- x(t-1) + A 2 x(t-1)- A 2 x(t-1) + A 2 x(t-2) = (A 1 +A 2 -I)x(t-1)-A 2 Δx(t-1), rewritten as Δx(t) = Ax(t-1) + BΔx(t-1) = ab x(t-1) + BΔx(t-1). If the coefficient matrix A of x(t-1) is zero, there are 3 unit roots since the relations are about the first-differences of the three variables. Otherwise write A as ab. If A is of rank 1, a is a column vector an b is a row vector which is the transpose of the cointegrating vector (a row vector of coefficients of the cointegrating equation b x). The cointegrating equation b x was estimated in section III by regressing log (P) on log (M2/Y), as suggested by Engle and Granger (1987) under our assumption that the coefficients of log (M2) and log(y) are opposite in sign and equal in magnitude. Note that the equilibrium relation between log(p), log(y) and log(m) estimated earlier in this lecture is a cointegration relation of a VAR. We apply the method of Johansen (1991) to find the rank of the coefficient matrix A of x(t-1). The result is than the rank is 1. We then use the above method to find the impulse responses of log(p) and log(y) to exogenous changes (or impulses) of log(m).

20 VAR and Impulse response To study the effect of monetary policy using such a model one cannot simply use the coefficients a3 and b3 in equation (6.8) which show the effects of m(t-1) on log price p and log output y respectively. The reason is that m itself is determined by its own lagged values and the lagged values of p and y. If the government is to exercise any monetary policy affecting m it has to be other than what the lagged variables on the righthand side of the equation for m in (6.8) would determine, i.e., it has to show up in the residual u(3t). Therefore to study the effect of monetary shocks (those factors other than the lagged variables already included in the right-hand side of (6.8) as represented by u(3t), we have to trace the effects of u(3t) on each variable at t, t+1,, t+k by solving the equations (6.8) forward given the initial values of p(t-1), y(t-1), m(t-1), p(t-2), y(t- 2) and m(t-2), There is a technical problem in attributing the effects of u3t on pt and yt since it does not appear in the first two equations of (6.8) but u(1t) and u(2t) in these equations may be correlated with it. Leaving this technical problem aside, we can solve equations (6.8) for t, t+1, and so forth. The effect of u(3t) on p(t+k) is the impulse response of p(t+k) to a shock (impulse) in m.

21 VAR in matrix notations and written as a first-order system VAR of higher order can be written as VAR of first order by redefining the vector of dependent variables: y t = A 1 y t-1 + A 2 y t-2 + A 3 y t-3 + v t can be written as y t A 1 A 2 A 3 y t-1 v 1t y t-1 = I 0 0 y t-2 + v 2t y t-2 0 I 0 y t-3 v 3t Redefining the column on the left-hand side of the above matrix equation as y t, the matrix of coefficients as A and the vector of residuals as v t we can write this third-order system as a first-order system y t = Ay t-1 + v t.

22 VAR and Impulse response in matrix notations To define the impulse response functions we use the VAR to substitute out past y t-k by past v t-k thus: y t = v t +Ay t-1 = v t + A(v t-1 +Ay t-2 ) = = v t + Av t-1 + A 2 v t A k v t-k + where A 3 = AAA which a square matrix of the same dimension as the matrix A. This shows that the effect of a past residual (shock or impulse) v t-k on the current y t is given by A k v t-k. For a one unit change in v t-k the current y t will be changed by A k units. Let A be a 3 by 3 matrix. The impulse response of y 2t to v 1, t-k is given by the 2-1 element of the matrix A k.

23 Figure 6.1 Impulse Responses of Price and Real Output to Chinese M1

24 Research topic Add to the above 3-variable VAR: 1. Δlog(exports) 2. Δlog(exports imports) 3. Δlog(exports) and Δlog(imports) separately. Observe the changes in the impulse response functions of logp and logy to logm. Use VAR for forecasting. Investigate the meaning of the statement that output growth is driven by growth in exports.