1. Introduction to
Macroeconometric Models |

1.1 Macroeconometric
Models |

A macroeconometric model like the US model is a set of equations designed to
explain the economy or some part of the economy. There are two types of equations: stochastic,
or behavioral, and identities. Stochastic equations are estimated from
the historical data. Identities are equations that hold by definition; they are always
true. There are two types of variables in macroeconometric models:
It is easiest to consider what a macroeconometric model is like by considering a
simple example. The following is a simple multiplier model. I is investment, _{t}Y is total income
or GDP, _{t}G is government spending, and _{t}r is the
interest rate. The _{t}t subscripts refer to period t. (1) C _{t} = a_{1} + a_{2}Y_{t} + e_{t}(2) I_{t} = b_{1} + b_{2}r_{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,
and _{t}Y; they are explained by the model. _{t}r and _{t}G
are exogenous variables; they are not explained. _{t}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
u 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 _{t}e
in equation (1) captures all the factors that affect consumption other than current
income. Likewise, the error term _{t}u in equation (2) captures all the
factors that affect investment other than the interest rate. _{t}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, 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).
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 a, _{2}b,
and _{1}b. 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. _{2}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 a
would be the intercept. The _{1}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 a, _{2}b, and _{1}b.
_{2}
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 I, and _{t}Y. Thus, the model can be solved to find
the three unknowns. _{t}A model like equations (1)-(3) is called 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 is solved, _{t}C can then be solved. In
general, however, models are not this simple, and in practice models are usually solved
numerically using the _{t}Gauss-Seidel technique. The steps of the Gauss-Seidel
technique are as follows: - Guess a set of values for the endogenous variables.
- Using this set of values for the right hand side variables, solve all the equations for the left hand side variables.
- 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.
- 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 C from equation (1) and _{t}I from equation
(2), and use the computed values to solve for _{t}Y in equation (3). This
new value of _{t}Y would then be used for the next pass through the
model. _{t}
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
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
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 G 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 _{t}G (and _{t}r), perhaps the
historical value of _{t}G if the value for period _{t}t is known.
Let Y be the solution value of _{t}^{*}Y.
Now change the value of _{t}G (but not _{t}r) and solve
the model for this new value. Let _{t}Y be this new solution
value. Then _{t}^{**}Y is the change
in income that has resulted from the change in government spending. (The change in _{t}^{**} - Y_{t}^{*}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, 2)
solving the model for a new value of _{t}r, 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. _{t}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.
Actual models are obviously more complicated than equations (1)--(3) above. For
one thing, C,
which is an endogenous variable. If _{t}C appeared as an explanatory
variable in equation (1), then the model would include a lagged endogenous variable. _{t-1}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 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 (4) e _{t} = p e_{t-1} + v_{t}where p is the first order serial correlation coefficient and v
is an error term that is not serially correlated. In the estimation of an equation one can
treat _{t}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. |

1.2 Data |

It is important in macroeconomics to have a good understanding of the data.
Macroeconomic data are available at many different intervals. Data on variables like
interest rates and stock prices are available daily; data on variables like the money
supply are available weekly; data on variables like unemployment, retail sales, and
industrial production are available monthly; and variables from the National Income and
Product Accounts and Flow of Funds Accounts are available quarterly. It is always
possible, of course, to create monthly variables from weekly or daily variables, quarterly
variables from monthly variables, and so on. 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 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 change in Y from period t-1
to period t is Y. The _{t} - Y_{t-1}percentage
change in Y from t-1 to t is (Y,
which is the change in _{t} - Y_{t-1})/Y_{t-1}Y divided by Y. If, for example, _{t-1}Y
is 100 and _{t-1}Y 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. _{t}The percentage change in a variable is also called the If Y is 100 and _{t-1}Y 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. _{t} |