Copyright(© MTS-2002GG): You are free to use and modify these slides for educational purposes, but please if you improve this material send us your new version. Forecasting I (some general forecasting issues) Gloria González-Rivera University of California, Riverside and Jesús Gonzalo U. Carlos III de Madrid Spring 2002 “There are two kind of forecasters: those who don´t know and those who don´t know they don´t know” John Kenneth Galbraith (1993)

Forecasts are made to guide decisions in a variety of fields. Operations planning and Control: Firms use forecasts to decide what to produce, when to produce and where to produce. Marketing: Pricing decisions, distribution path decisions, and advertising expenditure decisions all rely heavily on forecasts of responses of sales to different marketing schemes. Economics: The forecast of the major economic variables, such as GDP, unemployment, consumption, investment, the price level, and interest rates are used for governments to guide monetary and fiscal policy. Private firms use them for strategic planning, because economy-wide economic fluctuations typically have industry-level and firm-level effects. Financial speculation: Speculators in asset markets have an interest in forecasting asset returns (stock returns, interest rates, exchange rates,...). Such forecasts are made routinely. Are these forecasts successful??? Forecasting in Action

Financial risk management: Volatility forecasts are crucial for evaluating and insuring risks associated with asset portfolios. Volatility forecasts are also crucial for firms and investors who need to price assets such options and other derivatives. Capacity planning: Capacity planning decisions rely heavily on a variety of forecasts related both to product demand and supply. Business and government planning: Business and governments of all sorts must constantly plan and justify their expenditures. A major component of the budgeting process is the revenue forecast. Demography: Population forecasts are crucial for planning government expenditure on health care, infrastructure, social insurance, antipoverty programs, and so forth. Forecasting in Action (cont)

Think on any economic variable you want to forecast. What do you need? Loss Function: Symmetric or Asymmetric Forecast Object: A time series, an event, …etc. Forecast Statement: Point, Range or forecast density Forecast Horizon: Short, Medium or Long Information Set: Univariate or Multivariate Methods and Complexity: Model, Parsimony Principle, …etc. Basic Elements of Any Forecast

Once you have done your forecast, someone else can come with another forecast of the same variable. How do you compare these forecasts? Forecast Evaluation: Different measures of the forecast errors (see slide 13). IT IS IMPORTANT TO REALICE THAT EVERY FORECAST HAS AN ERROR. In general this error come from three different sources: Specification Error Approximation Error Estimation Error Basic Elements of Any Forecast (cont)

The regression model is an explicitly multivariate model, in which variables are explained and forecast on the basis of their own history and the histories of other, related variables. You have already studied regression models in your Econometric course, and very likely you have covered the forecasting issue. In the next slides we will review it. Forecasting with Regression Models:

A conditional forecasting model is one that can be used to produce forecasts for a variable of interest, conditional upon assumptions about other variables. With the regression model, our h-step ahead conditional forecast for y, given that the h-step value of x is is Assuming normality, we use the conditional density forecast, and from it we get conditional interval forecasts. We make the procedure operational by replacing unknown parameters with estimates. (a) Conditional Forecasting Models

Forecasts are subjetc to error. There are at least three sources of such error: Specification uncertainty: All models are wrong!!!! Innovation uncertainty: Future innovations are not known when the forecast is made. Parameter uncertainty: The coefficients that we use to produce forecasts are, of course, just estimates, and the estimates are subject to sampling variability. Q1: Which type of uncertainty is less important?? Conditional Forecasting Models (cont) Conditional Forecasting Models (cont)

When using a conditional forecasting model, simple calculation allow us to quantify both innovation and parameter uncertainty. Consider the following simple example: Suppose we want to predict y T+h at x T+h = x * T+h. Then Thus with corresponding error Thus, Conditional Forecasting Models (cont) Conditional Forecasting Models (cont)

In the latter expression, the first term accounts for parameter uncertainty, while the second accounts for the usual innovation uncertainty. Taken together we get an operational density forecast that accounts for parameter uncertainty: from which interval forecasts may be constructed as well. Conditional Forecasting Models (cont) Conditional Forecasting Models (cont)

Often we do not want to make forecasts of y conditional upon assumptions about x, rather, we just want the best possible forecast of y-an unconditional forecast. To get an unconditional forecast from a regression model, we often encounter the forecasting the right-hand- side variables problem. That is, to get an optimal unconditional point forecast for y, we cannot insert an arbitrary value for future x, rather, we need to insert the optimal point forecast, x T+h,T,which yields the unconditional forecast We usually don`t have such a forecast for x and the regression model at hand doesn’t help us. Assuming this variable follows and ARIMA representation, you will learn how to produce these forecasts in the next set of slides: FORECASTING II (b) Unconditional Forecasting Models

There are many ways of making forecasts, but all of them need the following common ingredients in order for success: (i)that there are regularities to capture (ii)that such regularities are informative about the future (iii) they are encapsulated in the selected forecasting method, and (iv) non-regularities and excluded. The main alternatives are (for some of them see Reading I): 1)Guessing 2)Extrapolation 3)Leading Indicators 4)Surveys 5)Time-Series Models 6)Econometric Models Evaluation of Forecasts

The most common overall accuracy measures are: mean squared error: root mean squared error mean absolute error where e t+h,t =y t+h -y t+h,t are the forecast errors. Evaluation of Forecasts (cont)

Suppose two competing forecasting procedures produce errors e t (1) and e t (2) for t=1,..., T. Then if expected squared error is to be the criterion, the procedure yielding the lower MSE over the sample period will be judged superior. How can we test MSE(1) = MSE(2) versus the opposite? Assume that the individual forecast errors are unbiased and not autocorrelated. Consider, now, the pair of random variables e t (1 )+ e t (2) and e t (1 )- e t (2). Now so the two expected expected squared errors, will be equal iff this pair of random variables is uncorrelated. Q2: Find an easy way of testing this hypothesis (Hint: use regression analysis). (a) Comparing Forecast Accuracy

Let f t (1) and f t (2) be two forecasts of y t with errors Consider now a combined forecast, taken to be a weighted average of the two individual forecasts, The forecast error is Forecast combination

Hence the error variance is This expression is minimized for the value of k given by and substituting in the top expression, the minimum achievable error variance is Note that, unless. If either equality holds, then the variance of the combined forecast is equal to the smaller of the two error variances. Forecast combination (cont)

P1: Show that P 2: Explain what happens with as  approaches to –1 or +1. Problems on Forecast combination