R: How to fit a time series model such as “Y(t) = αX + βY(t-1)”?

问题

How do I fit this model in R, step by step? My scope is to make a forecast for t+1.

Y(t) = αX(t) + βY(t-1)

• Y(t) <- years from 1900 to 2000.
• X <- a score measure from 0 to 100.
• Y(t-1) <- lagged value of order 1 for Y.

Thanks in advance.

回答1:

Your model is an AR(1) time series for y with covariate x. We can just use arima0 (no missing value) or arima (missing value allowed) from R base:

fit <- arima0(y, order = c(1, 0, 0), xreg = x)

Let's consider a small example:

set.seed(0)
x <- runif(100)
## intercept: 0.1
## slope of `x`: 1.2
## AR(1) with coefficient 0.5
y <- 0.1 + 1.2 * x + arima.sim(list(ar = 0.5), n = 100, sd = 0.2)

fit <- arima0(y, order = c(1, 0, 0), xreg = x)

#Call:
#arima0(x = y, order = c(1, 0, 0), xreg = x)
#
#Coefficients:
#         ar1  intercept    xreg
#      0.4639     0.0645  1.2139
#s.e.  0.0879     0.0448  0.0590
#
#sigma^2 estimated as 0.03046:  log likelihood = 32.55,  aic = -57.11

Note the estimate is consistent with our true model.

---

Thanks. How do I insert more covariates (x1,x2,etc.), just in case?

Have a look at ?arima0 (or ?arima):

xreg: Optionally, a vector or matrix of external regressors, which
      must have the same number of rows as ‘x’.

You can specify a model matrix via xreg. Suppose you have regressors x1, x2, x3, in a data frame dat, you can generate this model matrix via:

X <- model.matrix(~ x1 + x2 + x3, dat)

Then

fit <- arima0(y, order = c(1, 0, 0), xreg = X)

回答2:

Use the forecast package and use an ARIMAX function and specify the structure, it will be (1,0,0) in this case. The xreg argument will allow you to include additional covariates.

It should look something like this..

library(forecast)
fit <- Arima(y, order=c(1,0,0),xreg = x)

来源：https://stackoverflow.com/questions/40384774/r-how-to-fit-a-time-series-model-such-as-yt-%ce%b1x-%ce%b2yt-1