[R] How to use "lag"?

[Gabor Grothendieck]

Spencer Graves <spencer.graves <at> pdf.com> writes:

: 
: Is it possible to fit a lagged regression, "y[t]=b0+b1*x[t-1]+e", 
: using the function "lag"?  If so, how?  If not, of what use is the 
: function "lag"?  I get the same answer from y~x as y~lag(x), whether 
: using lm or arima.  I found it using y~c(NA, x[-length(x)])).  Consider 
: the following: 
: 
:  > set.seed(1)
:  > x <- rep(c(rep(0, 4), 9), len=9)
:  > y <- (rep(c(rep(0, 5), 9), len=9)+rnorm(9)) # y[t] = x[t-1]+e
:  >
:  > lm(y~x)
: (Intercept)            x 
:      1.2872      -0.1064 
:  > lm(y~lag(x))
: (Intercept)       lag(x) 
:      1.2872      -0.1064 
:  > arima(y, xreg=x)
:       intercept        x
:          1.2872  -0.1064
: s.e.     0.9009   0.3003
: sigma^2 estimated as 6.492:  log likelihood = -21.19,  aic = 48.38
:  > arima(y, xreg=lag(x))
:       intercept   lag(x)
:          1.2872  -0.1064
: s.e.     0.9009   0.3003
:  > arima(y, xreg=c(NA, x[-9]))
:       intercept  c(NA, x[-9])
:          0.4392        0.8600
: s.e.     0.2372        0.0745
: sigma^2 estimated as 0.3937:  log likelihood = -7.62,  aic = 21.25
:  > arima(ts(y), xreg=lag(ts(x)))
: arima(x = ts(y), xreg = lag(ts(x)))
:       intercept  lag(ts(x))
:          1.2872     -0.1064
: s.e.     0.9009      0.3003
: sigma^2 estimated as 6.492:  log likelihood = -21.19,  aic = 48.38
: 

Here is some sample code:

R> # following 3 lines are from your post
R> set.seed(1)
R> x <- rep(c(rep(0, 4), 9), len=9)
R> y <- (rep(c(rep(0, 5), 9), len=9)+rnorm(9)) # y[t] = x[t-1]+e
R> 
R> # here are some examples using ts class - first one uses no lag
R> lm(y ~ x, cbind(y = ts(y), x = ts(x)))

Call:
lm(formula = y ~ x, data = cbind(y = ts(y), x = ts(x)))

Coefficients:
(Intercept)            x  
     1.2872      -0.1064  

R> # now lets redo it with a lag. 
R> lm(y ~ lagx, cbind(y = ts(y), lagx = lag(ts(x), -1)) )

Call:
lm(formula = y ~ lagx, data = cbind(y = ts(y), lagx = lag(ts(x),     -1)))

Coefficients:
(Intercept)         lagx  
     0.4392       0.8600  

R> # here is arima without a lag
R> b <- cbind(ts(y), ts(x))
R> arima(b[,1], order = c(1,1,1), xreg = b[,2])

Call:
arima(x = b[, 1], order = c(1, 1, 1), xreg = b[, 2])

Coefficients:
         ar1      ma1   b[, 2]
      0.3906  -1.0000  -0.3803
s.e.  0.4890   0.4119   0.3753

sigma^2 estimated as 7.565:  log likelihood = -20.2,  aic = 48.4

R> # and now we redo arima with a lag
R> bb <- cbind(ts(y), lag(ts(x),-1))
R> arima(bb[,1], order = c(1,1,1), xreg = bb[,2])

Call:
arima(x = bb[, 1], order = c(1, 1, 1), xreg = bb[, 2])

Coefficients:
          ar1      ma1  bb[, 2]
      -0.2991  -0.8252   0.8537
s.e.   0.4516   1.0009   0.0838

sigma^2 estimated as 0.444:  log likelihood = -7.9,  aic = 23.8

R> # you can alternately use the I notation with lm and ts objects
R> # if you load zoo first
R> library(zoo)
R> yt <- ts(y); xt <- ts(x)
R> lm(I(yt ~ xt))

Call:
lm(formula = I(yt ~ xt))

Coefficients:
(Intercept)           xt  
     1.2872      -0.1064  

R> lm(I(yt ~ lag(xt, -1)))

Call:
lm(formula = I(yt ~ lag(xt, -1)))

Coefficients:
(Intercept)  lag(xt, -1)  
     0.4392       0.8600