Forecasting with MA(1) Models

Financial Time Series - Lecture 6

• Lecture is about Forecasting with MA models

Forecasting Theorem Review

• The optimal point forecast (m-step ahead) is its conditional mean.

Forecast with MA(1)

• The forecast error of the 1-step ahead forecast of MA(1) is calculated
• The variance of the 1-step ahead forecast error is given.
• If is normally distributed, then a 1-step ahead 95% interval forecast can be calculated.
• The forecast error of 2-step ahead forecast of MA(1) is calculated.
• The variance of the 2-step head forecast error is calculated
• The variance of the 2-step head forecast error is greater than the variance of the 1-step head forecast error
• If is normally distributed, then a 2-step ahead 95% interval forecast is calculated
• For an MA(1) model, the multi-step ahead forecasts will tend toward the mean after the first step.
• The forecast error of the m-step ahead forecast of MA(1) is calculated.
• The variance of the m-step head forecast error is calculated.
• The variance of the m-step head forecast error is the same as the variance of the 2-step head forecast error
• If is normally distributed, then an m-step ahead 95% interval forecast is calculated

Forecast with MA(2)

• The forecast error of the 1-step ahead forecast of MA(2) is calculated.
• The variance of the 1-step ahead forecast error is calculated.
• If is normally distributed, then a 1-step ahead 95% interval forecast is calculated.
• The forecast error of the 2-step ahead forecast of AR(1) is calculated.
• The variance of the 2-step head forecast error is calculated.
• The variance of the 2-step head forecast error is greater than the variance of the 1-step head forecast error.
• If is normally distributed, then a 2-step ahead 95% interval forecast is calculated
• For an MA(2) model, the multi-step ahead forecasts go to the mean after the first two steps.

Forecast with MA(q)

• For an MA(q) model, the multi-step ahead forecasts go to the mean after the first q steps

Summary of MA Model Point Forecast

• MA(1) model point forecasts go to the mean after the first-step ahead forecast.
• MA(2) model point forecasts go to the mean after the second-step ahead forecast.
• MA(q) model point forecasts go to the mean after the qth step ahead forecast.

Forecasting with ARMA Models

• Forecast with ARMA(1,1)
• The forecast error of the 1-step ahead forecast of ARMA(1,1) is calculated.
• The variance of the 1-step ahead forecast error is calculated.
• If is normally distributed, then a 1-step ahead 95% interval forecast is calculated
• The forecast error of 2-step ahead forecast of ARMA(1,1) is calculated.
• The variance of the 2-step head forecast error is calculated.
• The variance of the 2-step head forecast error is bigger than the variance of the 1-step head forecast error. The more steps ahead forecast, the less accurate
• If is normally distributed, then a 2-step ahead 95% interval forecast is calculated
• Multiple-step ahead forecasts of ARMA(1,1) are similar to those of AR(1), except that the 1-step ahead forecast adjusts for the MA(1) part.
• Starting from the 2-step ahead forecast, ARMA(1,1) and AR(1) are the same

Forecast with ARMA(2,1)

• Multiple-step ahead forecasts of ARMA(2,1) are similar to those of AR(2), except that the 1-step ahead forecast is adjusted for the MA(1) part.
• Starting from the 2-step ahead forecast, ARMA(2,1) and AR(2) are the same.

Forecast with ARMA(1,2)

• Multiple-step ahead forecasts of ARMA(1,2) are similar to those of AR(1), except that the 1-step and 2-step ahead forecasts are adjusted for a MA part.
• Starting from the 3-step ahead forecast, ARMA(1,2) and AR(1) are the same.

Forecast with ARMA(2,2)

• The multiple-step ahead forecasts of ARMA(2,2) are similar to those of AR(2), except that 1-step and 2-step: ahead forecasts are adjusted for the MA part.
• After 3-step ahead forecasts: ARMA(2,2) and AR(2) are the same.

Forecast with ARMA(k,q)

• The multiple-step ahead forecasts of ARMA(k,q) are similar to those of AR(k), except that first q-step ahead forecasts are adjusted for the MA(q) part.

Linear multivariate time series analysis

• Comparing to the linear univariate time series analysis, the linear multivariate time series analysis studies more than one variable.

Procedure of Linear Multivariate Time Series Analysis

• Test for cross autocorrelation
• Test for the seasonality of each variable and do a seasonal adjustment, if necessary.
• Test for the stationarity of each variable
• Select the Linear multivariate time series model, VAR or VEC
• Estimate the parameters of the selected model.
• Forecast the future value using the past values of multiple values.

Step 1: Test for Current And Lag Cross Correlation

• The purpose of testing for the cross-correlation is to determine if it is necessary to use a multivariate model
• Only two time series variables ( and ) are assumed for simplicity.
• If there exists some current and lag cross correlation between and , then is affected by the current value of or the past value of , or is affected by the current value of or the past value of . It is appropriate to study the two variables and together, and a multivariate time series model such as VAR or VEC should be used.
• Otherwise, an univariate time series model should be used to study and separately using univariate time series analysis

Lag Cross-correlation

• Is the lag correlation coefficient between and . If then is related to the past value.
• May not be = to

Current Cross-correlation

• , which is the current correlation coefficient between and at the same time t. If then is related to the current value, or vice versa
• The multivariate Ljung-Box statistics Q(m) (also called multivariate portmanteau test) is used tot test if there exist current and lag cross-correlation

Hypothesis

• Null: at least one of the above is not equal to 0
• Alt: rejected if the p-value is less than 5%
• There is a cross-correlation between and, so consider building the VAR or VEC multivariate time series model.
• If the p-value is greater than a significance level of 5%, there isn't cross-correlation between and. Therefore, there isn't a need to do multivariate time series analysis.

Steps

• Step 2: Check for seasonality of each variable
• Step 3: Test for stationarity of each variable
• Step 4: Select the linear multivariate time series model

Step 4: Selecting a Linear Multivariate Time Series Model

• If step 1 determines that multivariate time series models should be used, choose which multivariate model should be used; VAR model or VEC model. Stationary test results are important when deciding on a model.
• The VAR model is used for stationary variables
• The VAR models for non-stationary variables will result in a spurious regression
• Non-stationary variable is also called unit-root variable.
• If the non-stationary variable contains one unit root, (ie, can be transformed into a stationary variable with first-differencing), the variable is said to be integrated of order 1, which is denoted as I(1).
• If it contains two unit roots, the variable is said to be integrated of order 2, represented I(2).
• The integrated (non-stationary) variables can be cointegrated.
• If variables have one unit root or I(1) and are cointegrated, use a first differencing Vector Error Correction (VEC) model; otherwise, for non-cointegrated variables, use first differencing Vector Autoregression (VAR) model.
• If the variables have two unti roots or are I(2), take the first difference to reduce to the one unit root I(1) and repeat the above test to test for cointegration. Use a second differencing VEC model if variables are cointegrated, VAr otherwise.

Cointegration Test

• If the linear combination of two non-stationary variables is stationary, these two non-stationary variables are cointegrated and thus have long-team relationship.
• IIf 2 variables and are non stationary, but is stationary, them and are cointegrated. The vector (a, b) is called cointegration vector"
• Cointegration can also be tested by using Johansen cointegration test
• If non-stationary variables are cointegrated, the VEC model can be built

Vector Autoregression (VAR)

• Simplest VAR contains 2 time series variables and, and is called bivariate VAR model
• A bivariate VAR model consists of the 2 regression equations:
  • dependent variable is in 1 equation.
  • dependent variable is in the other equation
• A VAR with n variables consists of n regression equations, where each variable is a dependent in 1 equation.
• Is the effect of on, if holding constant. If = 0, then is not affected by and is only effects by its own last term through
• If = the effect of on, if holding constant. If , then is not affected by, and is only aaffected by its own lag term through
• If and, there is unidirectional relationship from to
• If and, there is unidirectional relationship from to
• If and, there is a feedback relationship between and
• If and, then the 2 time-series variables are uncoupled.
• The advantages to VAR are the ease of estimation and the usability is adequate in forecasting,. The lead-lag is a disadvantage , and an attempt to eliminate this is the introduction of structural form of VAR model.

This lecture discusses forecasting using Moving Average (MA) models, particularly MA(1). It covers calculating forecast errors and variances for 1-step and 2-step ahead forecasts. It explains how multi-step ahead forecasts tend toward the mean after the first step.