1.1 Macroeconometric Models
1.2 Data

There are two types of variables in macroeconometric models: endogenous and exogenous. Endogenous variables are explained by the equations, either the stochastic equations or the identities. Exogenous variables are not explained within the model. They are taken as given from the point of view of the model. For example, suppose you are trying to explain consumption of individuals in the United States. Consumption would be an endogenous variable-a variable you are trying to explain. One possible exogenous variable is the income tax rate. The income tax rate is set by the government, and if you are not interested in explaining government behavior, you would take the tax rate as exogenous.

Specification

It is easiest to consider what a macroeconometric model is like by considering a simple example. The following is a simple multiplier model. C_t is consumption, I_t is investment, Y_t is total income or GDP, G_t is government spending, and r_t is the interest rate. The t subscripts refer to period t.

(1) C_t = a₁ + a₂Y_t + e_t

(2) I_t = b₁ + b₂r_t + u_t

(3) Y_t = C_t + I_t + G_t

Equation (1) is the consumption function, equation (2) is the investment function, and equation (3) is the income identity. Equations (1) and (2) are stochastic equations, and equation (3) is an identity. The endogenous variables are C_t, I_t, and Y_t; they are explained by the model. r_t and G_t are exogenous variables; they are not explained.

The specification of stochastic equations is based on theory. Before we write down equations (1) and (2), we need to specify what factors we think affect consumption and investment in the economy. We decide these factors by using theories of consumption and investment. The theory behind equation (1) is simply that households decide how much to consume on the basis of their current income. The theory behind equation (2) is that firms decide how much to invest on the basis of the current interest rate. In equation (1) consumption is a function of income, and in equation (2) investment is a function of the interest rate. The theories behind these equations are obviously much too simple to be of much practical use, but they are useful for illustration. In practice it is important that we specify our equations on the basis of a plausible theory. For example, we could certainly specify that consumption was a function of the number of sunny days in period t, but this would not be sensible. There is no serious theory of household behavior behind this specification.

e_t and u_t are error terms. The error term in an equation encompasses all the other variables that have not been accounted for that help explain the endogenous variable. For example, in equation (1) the only variable that we have explicitly stated affects consumption is income. There are, of course, many other factors that are likely to affect consumption, such as the interest rate and wealth. There are many reasons that not all variables can be included in an equation. In some cases data on a relevant variable may not exist, and in other cases a relevant variable may not be known to the investigator. We summarize the effects of all of the left out variables by adding an error term to the equation. Thus, the error term e_t in equation (1) captures all the factors that affect consumption other than current income. Likewise, the error term u_t in equation (2) captures all the factors that affect investment other than the interest rate.

Now, suppose that we were perfectly correct in specifying that consumption is solely a function of income. That is, contrary to above discussion, suppose there were no other factors that have any influence on consumption except income. Then the error term, e_t, would equal zero. Although this is unrealistic, it is clear that one hopes that consumption in each period is mostly explained by income. This would mean that the other factors explaining consumption do not have a large effect, and so the error term for each period would be small. This means that the variance of the error term would be small. The smaller the variance, the more has been explained by the explanatory variables in the equation. The variance of an error term is an estimate of how much of the left hand side variable has not been explained. In macroeconomics, the variances are never zero; there are always factors that affect variables that are not captured by the stochastic equations.

Equation (3), the income identity, is true regardless of the theories one has for consumption and investment. Income is always equal to consumption plus investment plus government spending (we are ignoring exports and imports here).

Estimation

Once stochastic equations have been specified (written down), they must be estimated if they are to be used in a model. Theories do not tell us the size of coefficients like a₁, a₂, b₁, and b₂. These coefficients must be estimated using historical data. Given the data and the specification of the equations, the estimation techniques choose the values of the coefficients that best "fit" the data in some sense.

Consider the consumption equation above. One way to think of the best fit of this equation is to graph the observations on consumption and income, with consumption on the vertical axis and income on the horizontal axis. You can then think of the best fit as trying to find the equation of the line that is "closest to" the data points, where a₂ would be the slope of the line and a₁ would be the intercept. The ordinary least squares technique picks the line that minimizes the sum of the squared deviations of each observation to the line.

A common estimation technique in macroeconometrics is two-stage least squares, which is the technique used to estimate the US model. This technique is similar to the ordinary least squares technique except that it adjusts for certain statistical problems that arise when there are endogenous variables among the explanatory (right hand side) variables. In our current example, one would estimate the four coefficients a₁, a₂, b₁, and b₂.

Solution

Once a model has been specified and estimated, it is ready to be solved. By "solving" a model, we mean solving for the values of the endogenous variables given values for the exogenous variables. Remember that exogenous variables are not explained within the model. Say that we are in period t-1 and we want to use the model to forecast consumption, investment, and income for period t. We must first choose values of government spending and the interest rate for period t. Since t is in the future, we do not know for certain what government spending and the interest rate will be in period t, and so we must make our best guesses based on available information. (For example, we might use projected government budgets.) In other words, our model forecasts what consumption, investment, and income will be in period t if the values we chose for government spending and the interest rate in period t are correct. We must also choose values for the error terms for period t, which in most cases are taken to be zero. Given the exogenous variable values, the values of the error terms, and the coefficient estimates, equations (1), (2), and (3) are three equations in three unknowns, the three unknowns being the three endogenous variables, C_t, I_t, and Y_t. Thus, the model can be solved to find the three unknowns.

A model like equations (1)-(3) is called simultaneous. Income is an explanatory variable in the consumption function, and consumption is a variable in the income identity. One cannot calculate consumption from equation (1) unless income is known, and one cannot calculate income from equation (3) unless consumption is known. We thus say that consumption and income are "simultaneously" determined. (Investment in this model is not simultaneously determined because it can be calculated once the value for the exogenous variable r_t has been chosen.)

The above model, even though it is simultaneous, is easy to solve by simply substituting equations (1) and (2) into (3) and solving the resulting equation for Y_t. Once Y_t is solved, C_t can then be solved. In general, however, models are not this simple, and in practice models are usually solved numerically using the Gauss-Seidel technique. The steps of the Gauss-Seidel technique are as follows:

1. Guess a set of values for the endogenous variables.
2. Using this set of values for the right hand side variables, solve all the equations for the left hand side variables.
3. Step 2 yields a new set of values of the endogenous variables. Replace the initial set with this new set, and solve for the left hand side variables again.
4. Keep replacing the previous set of values with the new set until the differences between the new set and the previous set are within the required degree of accuracy. When the required accuracy has been reached, "convergence" has been attained, and the model is solved. The right hand side values are consistent with the computed left hand side values.

If a variable computed by an equation is used on the right hand side of an equation that follows, usually the newly computed value is used rather than the value from the previous iteration. It is not necessary to do this, but it usually speeds convergence. The usual procedure for the above model would be to guess a value for Y_t, compute C_t from equation (1) and I_t from equation (2), and use the computed values to solve for Y_t in equation (3). This new value of Y_t would then be used for the next pass through the model.

Testing

Once a model has been specified and estimated, it is ready to be tested. Testing alternative models is not easy, and this is one of the reasons there is so much disagreement in macroeconomics. The testing of models is discussed extensively in Fair (2003), Fair (1994), and Fair (1984), and the interested reader is referred to this material.

One obvious and popular way to test a model is to see how close its predicted values are to the actual values. Say that you want to know how well the model explained output and inflation in the 1970s. Given the actual values of the exogenous variables over this period, the model can be solved for the endogenous variables. The solution values of the endogenous variables are the predicted values. If the predicted values of output and inflation are close to the actual values, then we can say that the model did a good job in explaining output and inflation in the 1970s; otherwise not.

The solution of a model over a historical period, where the actual values of the exogenous variables are known, is called an ex post simulation. In this case, we do not have to guess values of the exogenous variables because all of these variables are known. One can thus use ex post simulations to test a model in the sense of examining how well it predicts historical episodes.

Forecasting

Once a model has been specified and estimated, it can be used to forecast the future. Forecasts into the future require that one first choose future values of the exogenous variables, as we described in the Solution section above. Given values of the exogenous variables, we can solve for the values of the endogenous variables. The solution of a model for a future period, where "guessed" values of the exogenous variables are used, is called an ex ante simulation.

Analyzing Properties of Models

Perhaps the most important use of a model is to try to learn about the properties of the economy by examining the properties of the model. If a model is an adequate representation of the economy, then its properties should be a good approximation to the actual properties of the economy. One may thus be able to use a model to get a good idea of the likely effects on the economy of various monetary and fiscal policy changes.

In the simple model above there are two basic questions that can be asked about its properties. One is how income changes when government spending changes, and the other is how income changes when the interest rate changes. In general one asks the question of how the endogenous variables change when one or more exogenous variables change. Remember, in our simple model above the only exogenous variables are government spending and the interest rate.

The Gauss-Seidel technique can be used to analyze a model's properties. Consider the question of how Y_t changes when G_t changes in the above model. In other words, one would like to know how income in the economy is affected when the government changes the amount that it spends. One first solves the model for a particular value of G_t (and r_t), perhaps the historical value of G_t if the value for period t is known. Let Y_t^* be the solution value of Y_t. Now change the value of G_t (but not r_t) and solve the model for this new value. Let Y_t^** be this new solution value. Then Y_t^** - Y_t^* is the change in income that has resulted from the change in government spending. (The change in Y divided by the change in G is sometimes called the "multiplier," hence the name of the model.) Similarly, one can examine how income changes when the interest rate changes by 1) solving the model for a given value of r_t, 2) solving the model for a new value of r_t, and 3) comparing the predicted values from the two solutions. You can begin to see how all sorts of proposed policies can be analyzed as to their likely effects if you have a good macroeconometric model.

Most of the experiments in this workbook are concerned with examining the properties of the US model. You will be comparing one set of solution values with another. If you understand these properties and if the model is an adequate approximation of the economy, then you will have a good understanding of how the economy works.

Further Complications

Actual models are obviously more complicated than equations (1)--(3) above. For one thing, lagged endogenous variables usually appear as explanatory variables in a model. The value of consumption in period t-1, denoted C_t-1, is a lagged endogenous variable since it is the lagged value of C_t, which is an endogenous variable. If C_t-1 appeared as an explanatory variable in equation (1), then the model would include a lagged endogenous variable.

When lagged endogenous variables are included in a model, the model is said to be dynamic. An important feature of a dynamic model is that the predicted values in one period affect the predicted values in future periods. What happens today affects what happens tomorrow, and the model is dynamic in this sense. Again, in the case of consumption, the idea is that how much you decide to consume this year will affect how much you decide to consume next year.

Also, most models in practice are nonlinear, contrary to the above model, where, for example, consumption is a linear function of income in equation (1). In particular, most models include ratios of variables and logarithms of variables. Equation (1), for example, might be specified in log terms:

(1)' log C_t = a₁ + a₂log Y_t + e_t

Nonlinear models are difficult or impossible to solve analytically, but they can usually be solved numerically using the Gauss-Seidel technique. The same kinds of experiments can thus be performed for nonlinear models as for linear models. As long as a model can be solved numerically, it does not really matter whether it is nonlinear or not for purposes of forecasting and policy analysis.

The error terms in many stochastic equations in macroeconomics appear to be correlated with their past values. In particular, many error terms appear to be first order serially correlated. If e_t is first order serially correlated, this means that:

(4) e_t = p e_t-1 + v_t

where p is the first order serial correlation coefficient and v_t is an error term that is not serially correlated. In the estimation of an equation one can treat p as a coefficient to be estimated and estimate it along with the other coefficients in the equation. This is done for a number of the stochastic equations in the US model.

The US model is a quarterly model; all the variables are quarterly. An important point should be kept in mind when dealing with quarterly variables. In most cases quarterly variables are quoted seasonally adjusted at annual rates. For example, in the National Income and Product Accounts real GDP for the fourth quarter of 1994 is listed as $7461.1 billion, but this does not mean that the U.S. economy produced $7461.1 billion worth of output in the fourth quarter. First, the figure is seasonally adjusted, which means that it is adjusted to account for the fact that on average more output is produced in the fourth quarter than it is in the other three quarters. The number before seasonal adjustment is higher than the seasonally adjusted number. Seasonally adjusting the data smooths out the ups and downs that occur because of seasonal factors.

Second, the figure of $7461.1 billion is also quoted at an annual rate, which means that it is four times larger than the amount of output actually produced (ignoring seasonal adjustment). Being quoted at an annual rate means that if the rate of output continued at the rate produced in the quarter for the whole year, the amount of output produced would be $7461.1 billion. For variables that are quoted at annual rates, it is not the case that the yearly amount is the sum of the four published quarterly amounts. The yearly amount is one fourth of this sum, since all the quarterly amounts are multiplied by four.

It is also important to understand how growth rates are computed. Consider a variable Y_t. The change in Y from period t-1 to period t is Y_t - Y_t-1. The percentage change in Y from t-1 to t is (Y_t - Y_t-1)/Y_t-1, which is the change in Y divided by Y_t-1. If, for example, Y_t-1 is 100 and Y_t is 101, the change is 1 and the percentage change is .01. The percentage change is usually quoted in percentage points, which in the present example means that .01 would be multiplied by 100 to make it 1.0 percent.

The percentage change in a variable is also called the growth rate of the variable, except that in most cases growth rates are given at annual rates. In the present example, the growth rate in Y at an annual rate in percentage points is

Growth Rate (annual rate) = 100[(Y_t/Y_t-1)⁴ - 1] .

If Y_t-1 is 100 and Y_t is 101, the growth rate is 4.06 percent. Note that this growth rate is slightly larger than four times the quarterly growth rate of 1.0 percent. The above formula "compounds" the growth rate, which makes it slightly larger than 4 percent. The growth rate at an annual rate is the rate that the economy would grow in a year if it continued to grow at the same rate in the next three quarters as it did in the current quarter.