Which time series models uses the forecast error in the previous period as part of calculating the forecast for the next subsequent time period?
2022 Curriculum CFA Program Level II Quantitative Methods Show IntroductionAs financial analysts, we often use time-series data to make investment decisions. A time series is a set of observations on a variable’s outcomes in different time periods: the quarterly sales for a particular company during the past five years, for example, or the daily returns on a traded security. In this reading, we explore the two chief uses of time-series models: to explain the past and to predict the future of a time series. We also discuss how to estimate time-series models, and we examine how a model describing a particular time series can change over time. The following two examples illustrate the kinds of questions we might want to ask about time series. Suppose it is the beginning of 2020 and we are managing a US-based investment portfolio that includes Swiss stocks. Because the value of this portfolio would decrease if the Swiss franc depreciates with respect to the dollar, and vice versa, holding all else constant, we are considering whether to hedge the portfolio’s exposure to changes in the value of the franc. To help us in making this decision, we decide to model the time series of the franc/dollar exchange rate. Exhibit 1 shows monthly data on the franc/dollar exchange rate. The data are monthly averages of daily exchange rates. Has the exchange rate been more stable since 1987 than it was in previous years? Has the exchange rate shown a long-term trend? How can we best use past exchange rates to predict future exchange rates? As another example, suppose it is the beginning of 2020. We cover retail stores for a sell-side firm and want to predict retail sales for the coming year. Exhibit 2 shows monthly data on US retail sales. The data are not seasonally adjusted, hence the spikes around the holiday season at the turn of each year. Because the reported sales in the stores’ financial statements are not seasonally adjusted, we model seasonally unadjusted retail sales. How can we model the trend in retail sales? How can we adjust for the extreme seasonality reflected in the peaks and troughs occurring at regular intervals? How can we best use past retail sales to predict future retail sales? Some fundamental questions arise in time-series analysis: How do we model trends? How do we predict the future value of a time series based on its past values? How do we model seasonality? How do we choose among time-series models? And how do we model changes in the variance of time series over time? We address each of these issues in this reading. We first describe typical challenges in applying the linear regression model to time-series data. We present linear and log-linear trend models, which describe, respectively, the value and the natural log of the value of a time series as a linear function of time. We then present autoregressive time-series models—which explain the current value of a time series in terms of one or more lagged values of the series. Such models are among the most commonly used in investments, and the section addresses many related concepts and issues. We then turn our attention to random walks. Because such time series are not covariance stationary, they cannot be modeled using autoregressive models unless they can be transformed into stationary series. We therefore explore appropriate transformations and tests of stationarity. The subsequent sections address moving-average time-series models and discuss the problem of seasonality in time series and how to address it. We also cover autoregressive moving-average models, a more complex alternative to autoregressive models. The last two topics are modeling changing variance of the error term in a time series and the consequences of regression of one time series on another when one or both time series may not be covariance stationary. Learning OutcomesThe member should be able to:
Summary
Which model uses past errors for forecasting?Rather than using past values of the forecast variable in a regression, a moving average model uses past forecast errors in a regression-like model.
What is forecasting error in time series?In statistics, a forecast error is the difference between the actual or real and the predicted or forecast value of a time series or any other phenomenon of interest.
Which model is used for time series forecasting?AutoRegressive Integrated Moving Average (ARIMA) models are among the most widely used time series forecasting techniques: In an Autoregressive model, the forecasts correspond to a linear combination of past values of the variable.
Which one of the following is a measure for forecasting error?Mean absolute deviation (MAD) is another commonly used forecasting metric. This metric shows how large an error, on average, you have in your forecast.
|