I have obtained different values of intercept from autoregressive AR(1) model calculated with `lm`

and `arima`

, but coefficients before lagged variable are almost exactly the same, intercept for `arima`

is `0.1153`

and `0.03014`

for `lm`

why does this difference occurs ?

`> set.seed(123) > ar1 <- arima.sim(n = 1000, list(ar = 0.8, sd = 1)) #simulated dataset > arima(ar1, order=c(1,0,0)) #AR(1) model Call: arima(x = ar1, order = c(1, 0, 0)) Coefficients: ar1 intercept 0.7772 0.1153 s.e. 0.0200 0.1418 sigma^2 estimated as 1.006: log likelihood = -1422.16, aic = 2850.33 > summary(lm(ar1[-1]~ar1[-1000])) Call: lm(formula = ar1[-1] ~ ar1[-1000]) Residuals: Min 1Q Median 3Q Max -2.8835 -0.6563 -0.0029 0.6587 3.1793 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.03014 0.03181 0.947 0.344 ar1[-1000] 0.77612 0.02000 38.802 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.003 on 997 degrees of freedom Multiple R-squared: 0.6016, Adjusted R-squared: 0.6012 F-statistic: 1506 on 1 and 997 DF, p-value: < 2.2e-16 `

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#### Best Answer

Stationary time series estimation first starts by centering. So the intercept estimation in `arima`

is just `mean(ar1)`

.

Turning to `lm`

. Did you not see that the intercept is not statistically different from 0, as `ar1[-1]`

and `ar1[-1000]`

have the same mean? That's what stationarity implies.

In summary, `arima`

here is doing real estimation for `ar1`

, while `lm`

is only estimating its auto-correlation. You are doing different things!

If this is still not that clear for you, let's just focus on the `lm`

usage in general.

`# true model: y = 0.7 x + 0.5 x <- runif(1000) y <- 0.5 + 0.7 * x + rnorm(1000, 0, 0.05) # a proper estimation lm(y ~ x) # now center x and y so they have the same mean: 0 # this only estimates slope x1 <- x - mean(x) y1 <- y - mean(y) lm(y1 ~ x1) `

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