2.5 EXAMPLES OF APPLICATIONS OF FORECASTS IN MACROECONOMICS AND FINANCE

2.5 EXAMPLES OF APPLICATIONS OF FORECASTS IN MACROECONOMICS AND FINANCE

Forecasts are of interest to economic agents only in so far as they can help improve their decisions, so it is useful to illustrate the importance of forecasts in the context of some simple economic decision problems. This section provides three such examples from economics and finance.

2.5.1 Central Bank’s Decision Problem

Consider a central bank with an objective of targeting inflation by means of a single policy instrument, $y_t.$ , which could be an interest rate such as the repo rate, i.e., the rate charged on collateralized loans. Svensson (1997) sets out a simple model in which the central bank’s loss function depends on the difference between the inflation rate $({\mathbf{y}}_t)$ and a target inflation rate $({\mathbf{y}}^{\ast })$ . Svensson shows that, conditional on having chosen a value for its instrument (the repo rate), the central bank’s decision problem reduces to that of choosing a forecast that minimizes the deviation from the target. Although the forecast does not enter directly into the central bank’s loss function, it does so indirectly because the actual rate of inflation (which is what the central bank really cares about) is affected by the bank’s choice of interest rate which in turn reflects the inflation forecast.

Specifically, the central bank is assumed to choose a sequence of interest rates $\{i_{\tau }{\}}_{\tau =t}^{\infty }$ to minimize a weighted sum of expected future losses,

$$E_t\sum _{\tau =t}^{\infty }{\lambda }^{\tau -t}L(y_{\tau }-y^{\ast }),\text{\hspace{1em}(2.31)}$$

where $\lambda \in (0,1)$ is a discount rate and $E_t[]$ denotes the conditional expectation given information available at time t. Both current and future deviations from target inflation affect the central bank’s loss.

Following Svensson’s analysis, suppose the central bank has quadratic loss

$$\begin{array}{r}L(y_{\tau }-y^{\ast })=\frac{1}{2}(y_{\tau }-y^{\ast }{)}^2.\end{array}\text{\hspace{1em}(2.32)}$$

Future inflation rates depend on the sequence of interest rates which are chosen to minimize expected future loss and hence satisfy the condition

$${\left\{i_{\tau }^{\ast }\right\}}_t^{\infty }=\arg \underset{{\left\{i_{\tau }\right\}}_t^{\infty }}{\min }\sum _{\tau =t}^{\infty }{\lambda }^{\tau -t}{E}_t\left[(y_{\tau }-y^{\ast }{)}^2\right].\text{\hspace{1em}(2.33)}$$

Complicating matters, inflation is not exogenous but is affected by the central bank’s actions. Solving (2.33) is therefore quite difficult since current and future interest rates can be expected to affect future inflation rates. Because inflation forecasts matter only in so far as they affect the central bank’s interest rate policy and hence future inflation, a model for the data-generating process for inflation is needed. Svensson proposes a tractable approach in which inflation and output are generated according to the equations

$$y_{t+1}=y_t+{\alpha }_1{z}_t+{ϵ}_{t+1},\text{\hspace{1em}(2.34)}$$

$$z_{t+1}={\beta }_1{z}_t-{\beta }_2(i_t-y_t)+{\eta }_{t+1},\text{\hspace{1em}(2.35)}$$

where $z_t$ is current output relative to its potential level, and all parameters are positive, i.e., ${\alpha }_1,{\beta }_1,{\beta }_2>0$ . The quantities ${ϵ}_{t+1}$ and ${\eta }_{t+1}$ are unpredictable shocks to inflation and output, respectively. The first equation expresses the change in inflation as a function of the lagged output, while the second equation shows that the real interest rate $(i_t-y_t)$ impacts output with a lag and also allows for autoregressive dynamics assuming ${\beta }_1<1$ . Using these equations to solve for inflation two periods ahead, we obtain the following equation:

$$y_{t+2}=(1+{\alpha }_1{\beta }_2)y_t+{\alpha }_1(1+{\beta }_1)z_t-{\alpha }_1{\beta }_2{i}_t+{ϵ}_{t+1}+{\alpha }_1{\eta }_{t+1}+{ϵ}_{t+2}.\text{\hspace{1em}(2.36)}$$

Notice that the policy instrument (i ) impacts the target variable $(\mathbf{y})$ with a two-period delay. Moreover, each interest rate affects one future inflation rate and so a solution to the infinite sum in (2.33) reduces to choosing $i_t$ to target $y_{t+2}.$ , choosing $i_{t+1}$ to target ${\gamma }_{t+3}.$ , etc. Hence, the central bank’s objective in setting the current interest

rate, $i_t,$ simplifies to

$$\underset{i_t}{\min }{E}_t\left[{\lambda }^2(y_{t+2}-y^{\ast }{)}^2\right].$$

Using the quadratic loss function in (2.32), the first-order condition becomes

$$E_t\left[\frac{\partial L(y_{t+2}-y^{\ast })}{\partial i_t}\right]=E_t\left[(y_{t+2}-y^{\ast })\frac{\partial y_{t+2}}{\partial i_t}\right]=0.\text{\hspace{1em}(2.37)}$$

From (2.36) this means choosing $i_t$ so that $E_t\left[y_{t+2}\right]=y^{\ast }$ , which can be accomplished by setting

$$i_t^{\ast }=y_t+\frac{(y_t-y^{\ast })+{\alpha }_1(1+{\beta }_1)z_t}{{\alpha }_1{\beta }_2}.\text{\hspace{1em}(2.38)}$$

It follows that the optimal current interest rate, $i_t^{\ast }$ , should be higher, the higher the current inflation rate as well as the higher the output relative to its potential, i.e., the lower the output gap.

Under this choice of interest rate level, the argument in the loss function reduces to

$$y_{t+2}-y^{\ast }=({ϵ}_{t+1}+{\alpha }_1{\eta }_{t+1}+{ϵ}_{t+2}).$$

This is just an example of certainty equivalence, which relies heavily on the chosen squared error loss function in (2.32). If the original loss function did not have a firstorder condition (2.37) that is linear in inflation, then the solution would not be so simple and the expected loss would not be a straightforward function of the expected inflation rate.