CODEWITHSUNDEEP
rtime-series
Gaurav Bansal
Gaurav Bansal

Reputation: 5660

Arima and lm not giving same coefficients in R

I'm fitting an Arima(2,0,0) model using the forecast package in R on the usconsumption dataset. However, when I mimic the same fit using lm, I get different coefficients. My understanding is that they should be the same. Below is my code.

> library(forecast)
> library(fpp)
> 
> #load data
> data("usconsumption")
> 
> #create equivalent data frame from time-series
> lagpad <- function(x, k=1) {
+     c(rep(NA, k), x)[1 : length(x)] 
+ }
> 
> usconsumpdf <- as.data.frame(usconsumption)
> usconsumpdf$consumptionLag1 <- lagpad(usconsumpdf$consumption)
> usconsumpdf$consumptionLag2 <- lagpad(usconsumpdf$consumption, 2)
> 
> #create arima and lm models
> arima1 <- Arima(usconsumption[,1], xreg=usconsumption[,2], order=c(2,0,0))
> lm1 <- lm(consumption~consumptionLag1+consumptionLag2+income, data=usconsumpdf)
> 
> #show coefficients
> arima1
Series: usconsumption[, 1] 
ARIMA(2,0,0) with non-zero mean 

Coefficients:
         ar1     ar2  intercept  usconsumption[, 2]
      0.1325  0.2924     0.5641              0.2578
s.e.  0.0826  0.0747     0.0883              0.0530

sigma^2 estimated as 0.3538:  log likelihood=-145.59
AIC=301.19   AICc=301.57   BIC=316.69
> summary(lm1)

Call:
lm(formula = consumption ~ consumptionLag1 + consumptionLag2 + 
    income, data = usconsumpdf)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.22400 -0.31689 -0.01079  0.34280  1.43839 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)      0.27373    0.08031   3.408 0.000829 ***
consumptionLag1  0.16423    0.07547   2.176 0.031039 *  
consumptionLag2  0.21857    0.07198   3.037 0.002800 ** 
income           0.26670    0.05247   5.082 1.04e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5952 on 158 degrees of freedom
  (2 observations deleted due to missingness)
Multiple R-squared:  0.2853,    Adjusted R-squared:  0.2717 
F-statistic: 21.02 on 3 and 158 DF,  p-value: 1.637e-11

Upvotes: 3

Views: 573

Answers (1)

Alex
Alex

Reputation: 4995

The documentation of arima() (Arima() is just a wrapper for arima()) tells this about the fitting method:

Fitting methods

The exact likelihood is computed via a state-space representation of the ARIMA process, and the innovations and their variance found by a Kalman filter. The initialization of the differenced ARMA process uses stationarity and is based on Gardner et al (1980). ...

Whereas lm uses Least Squares (QR-Factorization) as mentioned here: https://stats.stackexchange.com/questions/175983/whats-the-underlying-algorithm-used-by-rs-lm.

In the documentation I found this:

... an optional vector of weights to be used in the fitting process. If specified, weighted least squares is used with weights weights (that is, minimizing sum(w*e^2)); otherwise ordinary least squares is used.

Upvotes: 3

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