Autocorrelation When a regression model is fitted to time series - - PowerPoint PPT Presentation

ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

Autocorrelation

When a regression model is fitted to time series data, the residuals

• ften violate the standard assumptions by being correlated.

Because one residual is correlated with another in the same series, we refer to auto-correlation. Notation, emphasizing the autocorrelated nature of the residuals: Yt = E(Yt|xt) + Rt = β0 + β1xt,1 + β2xt,2 + · · · + βkxt,k + Rt.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

The correlation of Rt with Rt−1 is called a lagged correlation, specifically the lag 1 correlation. More generally, the correlation of Rt with Rt−m is the lag m correlation, m > 0. Autocorrelation is usually strongest for small lags and decays to zero for large lags. Autocorrelation depends on the lag m, but often does not depend

• n t.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

When the correlation of Rt with Rt−m does not depend on t, and in addition: the expected value E(Rt) is constant; the variance var(Rt) is constant; the residuals are said to be stationary. Regression residuals have zero expected value, hence constant expected value, if the model is correctly specified. They also have constant variance, perhaps after the data have been transformed appropriately, or if weighted least squares has been used.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

Correlogram

The graph of corr(Rt, Rt−m) against lag m is the correlogram, or auto-correlation function (ACF). For the SALES35 example:

sales <- read.table("Text/Exercises&Examples/SALES35.txt", header = TRUE) l <- lm(SALES ~ T, sales) acf(residuals(l))

The ACF is usually plotted including the lag 0 correlation, which is of course exactly 1. The blue lines indicate significance at α = .05.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

Modeling Autocorrelation

Sometimes the ACF appears to decay exponentially: ACF(m) ≈ φm, m ≥ 0, −1 < φ < 1. The 1st order autoregressive model, AR(1), has this property; if Rt = φRt−1 + ǫt, where ǫt has constant variance and is uncorrelated with ǫt−m for all m > 0, then corr(Rt, Rt−m) = φm, m > 0.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

General autoregressive model, AR(p): Rt = φ1Rt−1 + φ2Rt−2 + · · · + φpRt−p + ǫt. ACF decays exponentially, but not exactly as φm for any φ. Moving average models, MA(q): Rt = ǫt + θ1ǫt−1 + θ2ǫt−2 + · · · + θqǫt−q. ACF is zero for lag m > q.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

Combined ARMA(p, q) model: Rt = φ1Rt−1 + φ2Rt−2 + · · · + φpRt−p + ǫt + θ1ǫt−1 + θ2ǫt−2 + · · · + θqǫt−q. ACF again decays exponentially, but not exactly as φm for any φ.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

Regression with Autocorrelated Errors

To fit the regression model Yt = E(Yt|xt) + Rt we must specify both the regression part E(Yt|xt) = β0 + β1xt,1 + β2xt,2 + · · · + βkxt,k and a model for the autocorrelation of Rt, say ARMA(p, q) for specified p and q. Time series software fits both submodels simultaneously.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

In SAS: proc autoreg can handle AR(p) errors, but not ARMA(p, q) for q > 0. proc arima can handle ARMA(p, q). In R, try AR(1) errors:

ar1 <- arima(sales$SALES, order = c(1, 0, 0), xreg = sales$T) print(ar1) tsdiag(ar1)

The ARMA(p, q) model is specified by order = c(p, 0, q). The middle part of the order is d, for differencing.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

Note that ˆ φ, the ar1 coefficient, is significantly different from zero. Note that the trend (coefficient of T) is similar to the original regression: 4.2959 versus 4.2956. Its reported standard error is higher: 0.1760 versus 0.1069. The original, smaller, standard error is not credible, because it was calculated on the assumption that the residuals are not correlated. The new standard error recognizes autocorrelation, and is more credible, but is still calculated on an assumption: that the residuals are AR(1).

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

Which model to use? ACF of sales residuals decays something like an exponential, but is also not significantly different from zero for m > 1, so it could be MA(1).

ma1 <- arima(sales$SALES, order = c(0, 0, 1), xreg = sales$T) print(ma1) tsdiag(ma1)

ˆ θ1, the ma1 coefficient, is also significantly different from zero. But AIC is higher than for AR(1), so AR(1) is preferred.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

Be systematic; use BIC to find a good model:

P <- Q <- 0:3 BICtable <- matrix(NA, length(P), length(Q)) dimnames(BICtable) <- list(paste("p:", P), paste("q:", Q)) for (p in P) for (q in Q) { apq <- arima(sales$SALES, order = c(p, 0, q), xreg = sales$T) BICtable[p+1, q+1] <- AIC(apq, k = log(nrow(sales))) } BICtable

As usual, use k = log(nrow(...)) to get BIC instead of AIC. As in stepwise regression, minimizing AIC tends to overfit. AR(1) gives the minimum BIC out of these choices.

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ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II

Forecasting

with(sales, plot(T, SALES, xlim = c(0, 40), ylim = c(0, 200))) ar1Fit <- coefficients(ar1)["intercept"] + coefficients(ar1)["sales$T"] * sales$T lines(sales$T, ar1Fit, lty = 2) newTimes <- 36:40 p <- predict(ar1, n.ahead = length(newTimes), newxreg = newTimes) pCL <- p$se * qnorm(.975) matlines(newTimes, p$pred + cbind(0, -pCL, pCL), col = c("red", "blue", "blue"), lty = c(2, 3, 3))

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