Panel regression with ar1 model - r

Question

Excuse my naiveté. I'm not sure what this type of model is called -- perhaps panel regression.
Imagine I have the following data:
n <- 100
x1 <- rnorm(n)
y1 <- x1 * 0.5 + rnorm(n)/2
x2 <- rnorm(n)
y2 <- x2 * 0.5 + rnorm(n)/2
x3 <- rnorm(n)
y3 <- x3 * 0.25 + rnorm(n)/2
x4 <- rnorm(n)
y4 <- x4 * 0 + rnorm(n)/2
x5 <- rnorm(n)
y5 <- x5 * -0.25 + rnorm(n)/2
x6 <- rnorm(n)
y6 <- x6 * -0.5 + rnorm(n) + rnorm(n)/2
x7 <- rnorm(n)
y7 <- x7 * -0.75 + rnorm(n)/2
foo <- data.frame(s=rep(1:100,times=7),
y=c(y1,y2,y3,y4,y5,y6,y7),
x=c(x1,x2,x3,x4,x5,x6,x7),
i=rep(1:7,each=n))
Where y and x are individual AR1 time series measured over 100 seconds (I use 's' instead of 't' for the time variable) divided equally into groups (i). I wish to model these as:
y_t= b_0 + b_1(y_{t-1}) + b_2(x_{t}) + e_t
but while taking the group (i) into account:
y_{it)= b_0 + b_1(y_{it-1}) + b_2(x_{it}) + e_{it}
I wish to know if b_2 (the coef on x) is a good predictor of y and how that coef varies with group. I also want to know the R2 and RMSE by group and to predict y_i given x_i and i. The grouping variable can be discrete or continuous.
I gather that this type of problem is called panel regression but it is not a term that is familiar to me. Is using plm in R a good approach to investigate this problem?
Based on the comment below, I guess this is a simple start:
require(dplyr)
require(broom)
fitted_models <- foo %>% group_by(grp) %>% do(model = lm(y ~ x, data = .))
fitted_models %>% tidy(model)
fitted_models %>% glance(model)

Answer 1

Since you don't include fixed or random effects in the model, we are dealing with the pooled OLS (POLS) which can be estimated using lm or plm.
Let's construct example data of 100 groups and 100 observations for each:
df <- data.frame(x = rnorm(100 * 100), y = rnorm(100 * 100),
group = factor(rep(1:100, each = 100)))
df$ly <- unlist(tapply(df$y, df$group, function(x) c(NA, head(x, -1))))
head(df, 2)
# x y group ly
# 1 1.7893855 1.2694873 1 NA
# 2 0.8671304 -0.9538848 1 1.2694873
Then
m1 <- lm(y ~ ly + x:group, data = df)
is a model with a common autoregressive coefficient and a group-dependent effect of x:
head(coef(m1)[-1:-2], 5)
# x:group1 x:group2 x:group3 x:group4 x:group5
# -0.02057244 0.06779381 0.04628942 -0.11384630 0.06377069
This allows you to plot them, etc. I suppose one thing that you will want to do is to test whether those coefficients are equal. That can be done as follows:
m2 <- lm(y ~ ly + x, data = df)
library(lmtest)
lrtest(m1, m2)
# Likelihood ratio test
#
# Model 1: y ~ ly + x:group
# Model 2: y ~ ly + x
# #Df LogLik Df Chisq Pr(>Chisq)
# 1 103 -14093
# 2 4 -14148 -99 110.48 0.2024
Hence, we cannot reject that the effects of x are the same, as expected.