Chapter 9: Time Series Analysis

TIMESERIES ANALYSIS CHAPTER9 1 Autoregressive (AR) time-series models • Limitation of trend models: • When time-series residuals exhibit serial correlation, as evidenced by the DW test, we need to use the autoregressive (AR) model. We do this by regressing the dependent variable against one or more lagged values of itself, on condition that the modelled time-series is covariance stationary. • Conditions for covariance stationary: I. Constant and finite expected value (mean-reverting level). II. Constant and finite variance (homoscedastic). III. Constant and finite covariance between values at any given lag (the covariance of the time series with leading or lagged values of itself is constant). • An (AR) model of order p, AR(p) is expressed as: 𝒙𝒕 = 𝒃𝟎 + 𝒃 𝟏 𝒙 𝒕−𝟏 + 𝒃 𝟐 𝒙 𝒕−𝟐 + ⋯ + 𝒃𝒑𝒙𝒕−𝒑 + 𝜺 𝒕 • (Where p is the number of lagged values included as independent variables) EXHIBIT 12 Autoregression: AR(1) Model Gross Margin of Intel Quarterly Observations, January 2003–June 2019 Exhibit 12 shows the results of estimating the following AR(1) model: 𝑮𝒓𝒐𝒔𝒔 𝒎𝒂𝒓𝒈𝒊𝒏 𝒕 = 𝒃𝟎 + 𝒃 𝟏 𝑮𝒓𝒐𝒔𝒔𝒎𝒂𝒓𝒈𝒊𝒏𝒕−𝟏 + 𝜺𝒕 The estimated model is: 𝑮𝒓𝒐𝒔𝒔෣ 𝒎𝒂𝒓𝒈𝒊𝒏𝒕𝟑 𝟏 𝟓 𝟏 .𝟎 = 𝟏 − 𝒕 𝒏𝒊𝒈𝒓𝒂𝒎𝒔𝒔𝒐𝒓𝑮 𝟐 𝟔 𝟒 𝟕 .𝟎 + Source: Bloomberg Forecasting with an autoregressive model • When applying the chain rule of forecasting, it is necessary to calculate a one-step-ahead forecast before a two-step-ahead forecast. This implies that: o Multi-period forecasts are more uncertain than single-period forecasts. o Sample size = # of observations - # of (AR) order. • Detection and correction for autocorrelation in autoregressive models: • The DW test used with trend models is not appropriate with AR models. Instead, we follow the following steps to detect autocorrelation and make sure we correctly specify the AR model: I. Estimate the AR(1) model. II. Calculate the autocorrelation of the model’s residuals. III. Using a t-test, check whether the autocorrelations are significantly different from zero. With df=T-2, 𝒕 = IV. 𝑨𝒖𝒕𝒐𝒄𝒐𝒓𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏 𝑺𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝑬𝒓𝒓𝒐𝒓 = 𝝆𝜺 𝒕 ,𝜺 𝒕−𝒌 𝟏ൗ , where T is the number of observations. 𝑻 If any of the autocorrelations is significantly different from zero, we have not correctly specified the model. We need to add more lags to the model and repeat step II until all autocorrelations are insignificant. EXHIBIT 13 Monthly CPI Inflation at an Annual Rate: AR(1) Model—Monthly Observations, February 1995–December 2018 Exhibit 13 shows the results of estimating the model: 𝑰𝒏𝒇𝒍𝒂𝒕𝒊𝒐𝒏𝒕 = 𝒃𝟎 + 𝒃 𝟏 𝑰𝒏𝒇𝒍𝒂𝒕𝒊𝒐𝒏𝒕−𝟏 + 𝜺 𝒕 t = 1, 2, . . . , 359 ෣ 𝒕 𝑰𝒏𝒇𝒍𝒂𝒕𝒊𝒐𝒏 𝟔 𝟒 𝟑 𝟑 .𝟏 = 𝟏−𝒕𝒏𝒐𝒊𝒕𝒂𝒍𝒇𝒏𝑰 𝟒 𝟖 𝟗 𝟑 .𝟎 + Are these statistics valid? Becausethe lag 2 estimated autocorrelation has t-statistic larger than 1.97 (critical t-stat), the model is mis-specified. Residuals are serially correlated. Source: US Bureau of Labor Statistics. EXHIBIT14 Monthly CPI Inflation at an Annual Rate: AR(2) Model—Monthly Observations, March 1995–December 2018 Exhibit 14 shows the result of estimating a second time-series model, an AR(2) model using the same data as in the analysis shown in Exhibit 13. 𝑰𝒏𝒇𝒍𝒂𝒕𝒊𝒐𝒏𝒕 = 𝒃𝟎 + 𝒃 𝟏 𝑰𝒏𝒇𝒍𝒂𝒕𝒊𝒐𝒏𝒕−𝟏 + 𝒃 𝟐 𝑰𝒏𝒇𝒍𝒂𝒕𝒊𝒐𝒏𝒕−𝟐 + 𝜺 𝒕 The bottom portion of Exhibit 14 shows that none of the first four autocorrelations of the residual has a t-statistic greater in absolute value than the critical value of 1.97 and thus the model is now correctly specified. Source: US Bureau of Labor Statistics. Random walks and unit roots • Mean reversion and random walk: • For a time-series to be covariance stationary, it must have a constant and finite mean-reverting level, which is the value the time-series tends to move to. Once this level is reached, it is expected that 𝒙𝒕 = 𝒙𝒕−𝟏 for 𝑨𝑹 𝟏 ∴ 𝒙𝒕 = 𝒃𝟎 + 𝒃 𝟏 𝒙 𝒕−𝟏 ⟹𝒙𝒕 = 𝒃𝟎 + 𝒃 𝟏 𝒙 𝒕 ∴ mean-reverting level = 𝒕𝒙𝒃𝟎ൗ𝟏−𝒃𝟏 • For 𝑏1 = 1 (called unit root), the model does not have a finite meanreverting level, and thus, not covariance stationary. This happens when the model follows a random walk process. Random walks and unit roots • Unit root detection: • As testing if 𝑏1 = 1 cannot be performed directly, use Dickey and Fuller (DF) test by transforming the AR(1) model to run a simple regression by subtracting 𝑥 𝑡−1 from both sides as follows: 𝒙𝒕 − 𝒙𝒕−𝟏 = 𝒃𝟎 + 𝒃 𝟏 𝒙 𝒕−𝟏 − 𝒙𝒕−𝟏 ⟹𝒙𝒕 − 𝒙𝒕−𝟏 = 𝒃𝟎 + (𝒃 𝟏 −𝟏)𝒙 𝒕−𝟏 • Then test whether the transformed coefficient (𝑏 1 − 1) is different from zero using a modified t-test. With 𝑯𝟎: 𝒃 𝟏 − 𝟏 = 𝟎, if we fail to reject, we conclude that the series has a unit root. • Unit root correction: • Use first differencing to transform the data to a covariance stationary time series for which (𝑏1 – 1) ≠ 0. This is done by constructing the following AR(1) model: 𝒚𝒕 = 𝒃𝟎 + 𝒃 𝟏 𝒚 𝒕−𝟏 + 𝜺 where 𝑦 𝑡 = 𝑥𝑡 − 𝑥 𝑡−1 and 𝒚𝒕−𝟏 = 𝒙𝒕−𝟏 − 𝒙𝒕−𝟐 and 𝒃𝟎 = 𝒃 𝟏 = 𝟎 ⟹𝑥 = 𝑏0ൗ1−𝑏1 = 0Τ1−0( 0 = finite value) • If the data has a linear trend, first difference the data. If the data has an exponential trend, first difference the natural log of the data. EXHIBIT18 Yen/US Dollar Exchange Rate: AR(1) Model Month-End Observations, October 1980–August 2019 Exhibit 18 shows the results of estimating the model using month-end observations from October 1980 through August 2019. The results suggest that the yen/US dollar exchange rate is a random walk because the estimated intercept does not appear to be significantly different from 0 and the estimated coefficient on the first lag of the exchange rate is very close to 1. Source: US Federal Reserve Board of Governors. EXHIBIT 19 First-Differenced Yen/US Dollar Exchange Rate: AR(1) Model Month-End Observations, November 1980–August 2019 Exhibit 19 presents the results of the random walk model: 𝒚𝒕 = 𝒃𝟎 + 𝒃 𝟏 𝒚 𝒕−𝟏 + 𝜺 𝒕 Where: 𝒚𝒕 = 𝒙𝒕 − 𝒙𝒕−𝟏 Is the first-differenced exchange rate series and is covariance stationary. The results suggest that neither the intercept nor the coefficient on the first lag of the first-differenced exchange rate differs significantly from 0, and no residual autocorrelations differ significantly from 0. Source: US Federal Reserve Board of Governors.