Back Matter

ANNEX I: Model Structure and Identifying Assumptions

This annex discusses the general structure of the model estimated is section 4, and the number of restrictions required to identify the four unobserved shocks to demand and supply.

Following Blanchard-Quah (1989) and Judd and Trehan (1989) an underlying structural model is specified that relates the four endogenous variables (real GNP, employment, the real interest rate, and inflation) to the four unobserved shocks to labor supply, total factor productivity, and real and nominal aggregate demand. This model is described by equation (1) where Y denotes a (4x1) vector of endogenous variables, B(L) is an infinite-order matrix in the lag operator L, and S is a (4×1) vector of shocks.


The vector of shocks S appearing in this equation is assumed to follow the stochastic process given by equation (2), where U denotes a vector of innovations to these shocks and is assumed to be white noise.


By substituting equation (2) into (1), the structural model can be written in terms of the innovations to the shocks, U, rather than the shocks themselves (equation (3)). In equation (3), the matrix polynomial D(L) is a convolution of the intrinsic dynamics of the structural model as given by A(L), and the extrinsic dynamics of the shocks as given by B(L).


The structural model given by equation (3) is not directly observable and would require a large number of assumptions to be directly identified. The approach adopted in section 4 is to estimate a vector autoregression for Yt and use its moving-average representation to infer the structure of equation (3).

The estimated vector autoregression for Yt is given by equation (4), and its moving average representation is given in equation (5).


The identification of the unobserved vector of shocks U in equation (3) (strictly, the innovations in these shocks) can be achieved by equating the equation for Yt as given by (5) to the underlying structural model given by equation (3). This gives rise to:


where, on account of equality between the number of shocks and endogenous variables, the matrices M(L) and D(L) are dimensioned conformably. Equation (6) will hold for any set of residuals Vt that satisfies Vt = J Ut and M(L) = D(L) J-1. That is to say equation (6) will hold when written as


The identification of the four unobserved shocks reduces to determining the matrix J which satisfies equation (7). In general, with K variables and shocks, the matrix J is dimensioned K × K so a total of K × K identifying restrictions is required. A total of K × (K+1)/2 restrictions is provided by the assumption that the structural disturbances are uncorrelated and have unit variance. The remaining (K × K) - (K × (K+D/2) = K(K-1)/2) restrictions come from assumptions about the long-run effects of shocks as discussed in the text.

ANNEX II: Stationarity Properties of the Data

This annex examines the stationarity properties of the variables appearing in the vector autoregression estimated in section 4. Tests are recorded to determine whether each series is trend or difference stationarity, and whether the unit root in the process for real GNP is robust to a discrete change in the long-term growth rate of the economy. The data comprise the logarithms of real GNP, employment, prices, and the level of the (short-term) nominal interest rate. With appropriate transformations, these series encompass all the variables appearing in the vector autoregression of section 4. 34/

1. Trend versus difference stationarity

The tests for trend versus difference stationarity follow Perron, 35/ and are based on estimates of equations (1) through (2). In these equations y(t) denotes the tested variable, a(J) j = 1,5 denote parameters, T is a time trend, and N is the number of observations.


Equation (1) is used to test the null hypothesis that a series is a unit root process with drift (a1 ≠ 0, a2 = 1). equation (2) includes a time trend and is used to test for difference stationarity (a4 = 0, a5 = 1) versus trend stationarity (a4 ≠ 0, a5 < 1).

The tabulation below shows two different test statistics for equation (1): Z(ta2) is a t test for the null hypothesis that the series has a unit root (a2 = 1), Z(F) is an F test the joint hypothesis that the series is a unit root without drift (a2 = 1 and a1 = 0). All statistics are shown for two values of the Newey-West truncation parameter (h = 1 and 3). Under the null hypothesis of a unit root, the test statistics have non-standard distributions requiring critical values tabulated by Dickey and Fuller. Statistical significance at the 5 percent level is indicated by an asterisk.

Stationarity Tests

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The results from the tests recorded in the above tabulation can be summarized as follows. The null hypothesis of a unit root for the level of each series can not be rejected in any case. When prices are first differenced, the null hypothesis of a unit root in the inflation process is rejected, suggesting a unit root in the price level. All variables, with the exception of the nominal interest rate, appear to be characterized by drift.

Tests for difference versus trend stationarity are recorded in the following table.

Stationarity Tests

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This table shows two test statistics for equation (2): Z(ta5) is a t-test for the null hypothesis that a series has a unit root when a time trend is added to the equation: a5 = 1; Z(F) is a F-test for the joint hypothesis that a series has unit root and a coefficient on the time trend of zero: a4 = 0 and a5 = 1.

Based on the test statistics recorded in the above table the null hypothesis of a unit root in the level of each series can not be rejected. In addition, the joint hypothesis of a unit root and no time trend cannot be rejected for the level of any series.

2. Unit root in real GNP

This test examines whether the finding of a unit root in the process for real GNP is picking up the effects of a once-for-all decline in the growth rate of GNP in the late 1960s or early 1970s. 36/ The test, which follows Perron, 37/ is based on the residuals from equation (3).


where the coefficients a7 and a8 refer to two time trends, T1 and T2, included to capture a break in the growth rate of real GNP at some point in the sample. The augmented Dickey-Fuller test is applied to determine whether the residuals form this equation have a unit root (see equation (4)).


The table below summarizes test results for a unit root in the residuals of equation (3), a9 = 1 in equation (4), under two alternative assumptions about a break in trend growth. The statistics in the first row are for a break in 1973:1; those in the second, are for a break in 1969:1. (k refers to the number of lags added to equation (4) for the augmented Dickey-Fuller tests). Also shown are the estimates of a9 in equation (4).

Break in Real GNP Trend

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The table shows that the null hypothesis of a unit root in real GNP can not be rejected, whether the break in trend is assumed to occur in 1973 or 1969.


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The author would like to thank Bankim Chadha, Liam Ebrill, Owen Evans, and Ellen Nedde for helpful comments on an earlier draft.


For a discussion of the role of aggregate demand and supply shocks in the industrial countries during the 1970s, see Bruno and Sachs (1985). In addition, see Sheffrin (1989) for an interesting discussion on the role of supply and demand shocks in generating fluctuations in real GNP in the United States.


See, for example, Bruno and Sachs (1985).


For a balanced discussion of these models, see Sheffrin (1989).


The lag operator L is defined such that Ln xt = xt-n.


Stationarity of a variable in the weak sense implies that its unconditional mean and variance do not vary over time. See Stock and Watson (1988).


This assumption is consistent with most standard macroeconomic models. See, for example, Dornbusch and Fischer (1981) and Blanchard and Fischer (1989). Of course, the assumption is unlikely to be literally correct—shifts in investment will influence demand in the short run and supply in the long run—but will be a useful approximation provided there is not a problem of hysteresis. See Durlauf (1989).


Blanchard and Quah (1989) recognized that their approach would incorrectly identify temporary supply shocks as demand shocks. They were, however, unable to distinguish between these shocks without imposing restrictions on the short-run behavior of their model.


In order to ensure stationarity of the unemployment rate, Blanchard and Quah (1989) allowed for a discrete shift in the natural unemployment rate caused by demographic factors.


In addition, Shapiro and Watson (1988) allowed for oil price shocks. These shocks, however, played a very limited role in their model.


For further discussion of this point, see Hall (1988). Like Shapiro and Watson, we use manhours as our measure of labor supply but find that this does not lead to a dominant role for supply shocks.


Identification of these shocks would have required the assumption that the real interest rate was not affected by the rate of inflation in the long run.


Given that inflation is found to be stationary, the second identifying assumption is that the real interest rate is independent of the long-run price level. This is a less controversial assumption than that of independence of the inflation rate and the real interest rate.


For a discussion of the recent growth experience and the slowdown in the 1970s, see Denison (1979).


Stability of the capital-output ratio over long periods is one of the stylized features of the growth experience. See Solow (1957) and Denison (1979).


In writing down equations (7)’ and (8)’, and the equations that follow, constant terms are suppressed.


The finding that the real interest rate is nonstationary differs from Shapiro and Watson (1988). Given that inflation appears to be stationary, the nonstationarity in the real rate reflects nonstationarity in the nominal interest rate. For further discussion of the stochastic properties of real interest rates, see Rose (1988).


The number of restrictions required to exactly identify the model is equal to the square of the number of underlying shocks. We do not impose overidentifying restrictions so the restrictions used to identify the model cannot be tested (see Annex I).


This restriction implies that the real interest rate is not influenced by the price level in the long run.


Based on a specification search, six lags were used in the estimated equations.


We experimented with a number of different measures of the real interest rate. These included: the (ex post) real interest rate, which under rational expectations should differ from the ex ante real interest rate by an error that is orthogonal to agents’ information sets; and the real interest rate derived using the models’ predictions of inflation. In practice, the ex post real interest rate seemed to work best and was used in the estimations.


For further discussion of this point see Hall (1988).


Judd and Trehan (1989) were unable to reject the null hypothesis of a unit root in the nominal interest rate. Our finding that the inflation rate is stationary implies, under the maintained assumption of rational expectations, that the real and nominal interest rate contain the same unit root.


Had we found cointegrating relationships, we would have had to modify the vector autoregression to include the levels of cointegrated variables. The system would then have corresponded to a vector error-correction model (VECM).


Allowance was made for a slowdown in either the late 1960s or early 1970s. See Denison (1979) for a discussion of the timing of the growth slowdown in the United States.


With the autoregressive representation of the system written in matrix notation as


The contribution of a variable x to explaining the errors in forecasting y will depend on the estimated coefficients in the moving-average representation for y and the variance of x. With the moving average representation of the system given by


where B and S0 denote appropriately dimensioned matrices of coefficients, and XtB is the deterministic part of the system, the k-period ahead forecast error is given by:


Contrary to what was expected, the allowance for the growth slowdown in the early 1970s only increased the contribution of demand shocks to the unexplained variance of real GNP over a one- to eight-quarter horizon by between 5 and 15 percentage points.


This result is consistent with Blanchard and Quah’s (1989) finding that the unexplained innovation in the unemployment rate was strongly influenced by demand shocks in the short run.


This result was found to be sensitive to the number of lagged instruments used in estimating the real interest rate equation.


There is some evidence of a hump-shaped pattern in the response to supply shocks (see Blanchard and Quah (1989)).


In the neoclassical theory of growth, increases in labor supply might be expected to raise the real interest rate in the short run but have no effect in the long run. See Solow (1957).


The inflation rate is obtained by taking the first difference of the logarithm of prices; the (ex post) real interest rate is equal to the nominal interest rate in period t less the first difference of the logarithm of prices from period t to t+1. Under the assumption of rational expectations, the stationarity properties of the ex post and ex ante real interest rates coincide.


The discussion in the text explains why these periods were chosen.

Trends and Cycles in the U.S. Economy
Author: International Monetary Fund