396 PART 2: MULTI-EQUATION SIMULATION MODELS deviation of this sample distribution, which will give us an estimated value for the standard error of the forecast. We can then use this standard error of forecast just as we did in Chapter 8 to determine forecast confidence intervals. If the model is linear, then as the number of simulations included in the sample becomes large, the sample mean will approach the deterministic forecast (the forecast corresponding to all the random parameters set at 0). If the model is nonlinear, however, there is no guarantee that the sample mean will approach the deterministic forecast as the sample size increases,, and in fact it usually will not. In addition, it may be necessary for an unacceptably large number of simulations to be performed before the sample means for each variable converge at all. We therefore center our confidence intervals on the deterministic forecast rather than on the sample mean of the stochastic simulations. The process would be exactly the same if we wanted to forecast over a time horizon longer than one period. For each simulation we simply select a different random value for e1(, e2t, and e^, for each period, but we use the same random value for vn, v12, etc., during the entire simulation (since the equations of the model were specified and estimated under the assumption that the coefficients are constant over time). Furthermore, if the future values of the exogenous variables G, and M, were not known with certainty but had to be forecasted themselves, standard errors could be associated with their forecasts. The exogenous variables could then be treated as normally distributed random variables (with means equal to their forecasted values and standard deviations equal to their standard errors of forecast) in our stochastic simulation. For example, Eq. (13.40) could be rewritten to include an error term associated with G, (which now must itself be forecasted): Yt = C, + It + (G, + t)t) (13.41) Here G, is a forecast of G,, and tj, is a normally distributed random variable (defined for each period ŕ) with mean 0 and standard deviation equal to the standard error of the forecast G,. Note that our forecast is now a conditional forecast. APPENDIX 13.1 A Small Macroeconomic Model As a more detailed example of a simulation model, we estimate and simulate a small model of the United States economy.11 The model is intended to be illustrative and is therefore simple and highly aggregated, representing the economy in a way that is consistent with most introductory Keynesian macroeconomics textbooks. It does not represent the state of the art of macroeconomic modeling 11 Our thanks to John Simpson and Randall Mams for their help in the development of this model. CHAPTER 13: DYNAMIC BEHAVIOR OF SIMULATION MODELS 397 TABLE A13.1 VARIABLES IN THE MODEL Variables Equation number Endogenous: C Consumption (A13.4) GNP Gross national product (A13.1) UN Inventory investment (A13.7) INF Inflation rate (A13.10) INR Nonresidential investment (A13.5) IR Residential investment (A13.6) RL Long-term interest rate (A13.9) RS Short-term interest rate (A13.8) TAX Tax revenue (A13.3) UR Unemployment rate (A13.12) WINF Rate of growth of wage rate (A13.11) YD Disposable income (A13.2) Exogenous: G Government spending GNPP Potential GNP RGM Growth rate of the money supply POIL Growth rate of oil price TR Transfer payments WLTH Household wealth XM Net exports and forecasting, but it should be a good starting point for students interested in the development and use of multi-equation models. Model Specification and Estimation The model contains 12 equations, of which 9 are behavioral, 2 are identities (for GNP and disposable income), and the last estimates a relationship between taxes and GNP. Each equation in the multi-equation model is estimated using some form of two-stage least squares. In particular, when there are lagged dependent variables as well as serial correlation in the error terms. Fair's method is used for estimation. (This method is discussed in Chapter II.) We will describe the model and explain the theory underlying the specification on an equation-by-equation basis. A list of all the endogenous and exogenous variables of the model (with equation references) is given in Table A13.1.12 GNP and its components, consumption C, total investment /, government spending G, and net exports (exports minus imports) XM, are all in real (1982 dollar) terms. Total investment is disaggregated, and separate equations are estimated to explain nonresidential investment (fixed plant and equipment) 12 All the data series are listed in the instructor's manual that accompanies this text. All data are seasonally adjusted when appropriate. 398 PART 2: MULTI-EQUATION SIMULATION MODELS INR, residential investment IR, and investment in business inventories UN. Thus the GNP identity is written as GNP, = C, + INR, + IR, + UN, + G, + XM, (A13.1) Disposable income is given by YD, = GNP, - TAX, + TR, (A13.2) where TAX is the total tax flow, and TR is transfer payments. The tax mechanism in the model is extremely simple; the total tax flow is estimated as TAX, = -137.074 + .272GNP, (A13.3) (-7.54) (41.29) R2 = .995 s = 12.087 DW = 2.203 p = .748 Let us now turn to the consumption function. As one would expect, consumption can be largely explained by disposable income. However, we would expect that the marginal propensity to consume is higher for income received from transfer payments than for the rest of disposable income. (The majority of transfer payments go to lower-income individuals who have a higher marginal propensity to consume.) We therefore separate these two components of disposable income. In addition, we expect consumption to depend on household wealth WITH, as well as the short-term interest rate RS. (Higher interest rates increase the return from saving and also raise the cost of borrowing for large expenditures, thereby reducing consumption.) Our estimated equation for consumption is C, = -11.161 + .155(YD, - TR,) + .262TR, + 9.716WLTH, (-1.01) (4.94) (5.15) (3.40) - 2.590RS, + .739C,-! (A13.4) (-5.06) (15.34) R2 = .999 s = 10.839 DW = 1.86 Note that the estimated propensity to consume out of transfer payments is indeed considerably larger than out of the remainder of disposable income. Also, household wealth has a significant and positive impact on consumption, and the interest rate has a significant and negative impact. We expect nonresidential investment to depend positively on aggregate economic activity, and negatively on long-term interest rates. Our estimated equation is CHAPTER 13: DYNAMIC BEHAVIOR OF SIMULATION MODELS 399 INR, = -22.418 + .037GNP, - 1.174RL,_4 + .776INR,-i (A13.5) (-5.18) (6.24) (-2.82) (18.68) R2 = .994 s = 7.186 DW = 1.237 Residential investment is related to disposable income and the short-term interest rate (which serves as a proxy for the mortgage rate). Our estimated equation is IR, = -23.046 + .069YD, - 1.005RS,_! (Al3.6) (-.47) (3.85) (-2.11) R2 = .960 s = 6.735 DW = 1.209 p = .952 Inventory investment should depend on changes in output relative to changes in consumption. (Inventories accumulate when unanticipated increases in GNP exceed unanticipated increases in consumption.) The estimated equation is UN, = 3.582 + .509AGNP, - .647AC, + .726HN,_i (A13.7) (2.25) (10.97) (-6.39) (15.73) R2 = .770 s = 9.374 DW = 2.234 Note that both the change in GNP and the change in consumption have the correct signs and are highly significant. Our equation for the short-term interest rate represents a standard LM curve. The demand for money increases when disposable income increases, but decreases when the interest rate (the price of holding money) rises. Hence the real (net of inflation) interest rate rises with disposable income and falls with increases in the money supply. The nominal interest rate is the real rate plus the expected rate of inflation. We therefore estimate the following equation: RS, = -.805 + .002YD, - .116(AM,/M,_,) (-.49) (3.51) (-4.38) + .105(1^,-! + INF,_2 + INF,_3) (A13.8) (2.58) R2 = .856 s = 1.166 DW = 2.371 p = .745 Note that we use a moving average of inflation over the past three quarters as a proxy for expected future inflation. All three explanatory variables are statistically significant and have the expected signs. 400 PART 2: MULTI-EQUATION SIMULATION MODELS The long-term interest rate responds with a geometric lag to the short-term rate: RL, = .249 + .214RS, + .793RL,-! (Al 3.9) (2.21) (7.03) (26.89) R2 ■= .975 s = .475 DW = 2.103 We also estimate equations for the rates of price and wage inflation. We expect each of these variables to depend positively on the other. In addition, price inflation should depend on disposable income (reflecting demand pressure) as well as past rates of inflation. Finally, we introduce rate of growth of the price of oil as an exogenous variable; oil price inflation has been an important determinant of inflation in the United States and elsewhere. Our estimated equation is thus INF, = -1.231 + .521WINF, + .001YD, + .011POIL, (-2.03) (5.58) (2.21) (2.26) + .187(11^,-! + TNF,-2) (A13.10) (3.95) R2 = .723 5 = 1.540 DW = 2.294 In addition to price inflation, we expect the rate of wage inflation to depend on the unemployment rate. (A higher unemployment rate should reduce the bargaining power of workers for higher wage increases.) Our estimated equation is WINF, = 1.793 + .795INF, - .039UR,-3 (A13.il) (2.30) (11.36) (-.34) R2 = .537 s = 2.047 DW = 2.054 Note that the unemployment rate is statistically insignificant in this equation. Finally, an equation is estimated for the unemployment rate (measured as a percent). This variable is related to the change in disposable income (and thus output) and to the difference between actual and potential GNP. TJR, = 6.394 + .003AYD, - .013(GNP, - GNPP,) (A13.12) (14.39) (2.95) (-10.12) R2 = .967 s = .296 DW = 1.833 p = .938 Most of the explanation in this equation comes from the difference between actual and potential GNP. (The change in disposable income, however, has the wrong sign.) CHAPTER 13: DYNAMIC BEHAVIOR OF SIMULATION MODELS 401 TABLE A13.2 RESULTS OF SIMULATIONS 1957-4 to 1987-4 rms rms % error error 1979-1 to 1987-4 1986-1 rms error to 1987-4 Variable rms error rms % error rms % error C 16.541 .0101 17.979 .0082 55.84 .0224 INR 15.056 .0471 19.607 .0456 41.06 .0929 TAX 17.922 .0337 19.626 .0246 17.663 .0195 IR 20.759 .173 32.238 .275 8.255 .0412 UN 11.91 11.661 13.453 21.12 11.825 50.930 RS 1.688 .304 2.684 .2728 3.050 .534 RL .704 .0963 .960 .0935 1.180 .139 INF 1.508 .174 1.173 .564 1.845 1.061 WINF 2.021 .439 1.798 2.019 2.765 4.554 UR .890 .175 .715 .0922 .596 .0987 GNP 37.828 .0138 3,414.4 .998 89.128 .0233 Simulation of the Model The model can now be simulated as a complete system. Two historical simulations and an ex post forecast were performed in order to evaluate the model's ability to replicate the actual data.13 The first simulation covered the entire estimation period (1957-4 to 1987-4). The second runs from 1979-1 to 1987-4. Finally, to perform an ex post forecast, the model was reestimated using data from 1957-4 through only 1985-4, and was then simulated from 1986-1 to 1987-4. The results of these simulations are summarized in Table Al3.2. For each simulation, rms errors and rms percent errors are shown for all the endogenous variables. In addition, actual and simulated values for the endogenous variables are plotted in Figs. Al3.1 to Al3.9 for the historical simulation that covers the entire estimation period, and in Figs. A13.10toA13.12for the ex post forecast. The model reproduces the general movements of most variables but does not fully capture many of the sharp fluctuations that occurred over the business cycle. For example, the general movement of nonresidential investment is captured, but the extent of the downturns that occurred during the 1975 and 1982 recessions is not reproduced. And the model fails almost completely to capture the sharp fluctuations that occurred in residential investment. Fortunately, the historical simulation reproduces much of the movement in inventory investment, wage and price inflation, and the unemployment rate. (However, the ex post forecast of unemployment misses the drop that occurred in 1986-1987.) Finally, although it does not reproduce the sharp increases that occurred in " All simulations are dynamic, in the sense that simulated (rather than actual) values for the endogenous variables in a given period are used as inputs when the model is solved in future periods.