Is this a valid model specification? I am working with weekly, highly seasonal data (period=52) and fitting the following regression using lm:
$$
y_t=\rho_1y_{t-1}+\rho_2y_{t-52}+\beta x_t+\varepsilon_t
$$
where $x_t$ is a vector of independent variables that are also highly seasonal.
This isn't exactly a type of model I have seen before but it produces a good fit ($R^2=.8$) and fairly accurate forecasts. Is there a reason why this would not work or is there a better way to model this?(I am aware that working with weekly data like this is difficult because a year does not have exactly 52 weeks but since the seasonality is annual, it should not be a huge issue.)

I think the seasonality in $x_t$ should be accounted for, as otherwise it would tend to make the residuals seasonal. You could seasonally adjust $x_t$ before putting it into the model or include some seasonal terms (e.g. Fourier terms) in the model.
Side note: 52 vs. 52.18 weeks a year should not be a big issue for 10 or 20 years, but if your time span is much longer than that, it could make a difference.
