# Regression model

I'm analysing returns of funds and i have data on the following for 1000 funds.

• Returns in Period t
• Returns in Period t-1

I want to use a time series model to see how much returns have increased from period 1 to 2(b_1 in the following model). The model i had in mind is the following: $$y_t = b_0 + b_1(x_t) + e_t$$ Where $y_t$ is the returns in period $t$, and $x_t$ is the returns in period $t-1$.

I've been researching time series models lately and I found that if you have data like I do I need to find the:

Autoregressive order

Integrated order

Moving Average order

But can't I just see directly that my model is a AR(1) model? Since I'm only using 1 lagged variable? Or do i have to look at graphs to determine the orders?

Thank you

• If you know that that is your true model, then your AR order is 1, your Integrated order is 0 (you don't difference returns), and your moving average order is 0. You don't have to perform model seletion if you already have a very good idea what your model is. And yes, in this case you can use least-squared/regression to fit this model, estimating $b_0$, $b_1$ and the variance of your error. – Taylor Apr 26 '17 at 17:02
• @Taylor Thank you for your answer. Well, since i only have data on return in period t and t+1 for each fund. Doesn't that mean that it is a AR(1) Model? Or can it be a ARMA or ARIMA model aswell? – user358065 Apr 26 '17 at 17:06
• So your saying i could use a simple linear regression to estimate b_1? Will it not be unbiased since y_t and x_t is highly correlated? – user358065 Apr 26 '17 at 17:12
• @Taylor So your saying i could use a simple linear regression to estimate b_1? Will it not be unbiased since y_t and x_t is highly correlated? – user358065 Apr 26 '17 at 18:48
• An AR(1) is an ARMA(1,0) is an ARIMA(1,0,0) is the regression you wrote down. Your estimate for $b_1$ will not be unbiased. And it is precisely because $y_t$ and $x_t$ are correlated that $b_0 \neq 0$. – Taylor Apr 26 '17 at 19:07

The most commonly used technique to evaluate the parameters $p,q$ for $ARMA(p,q)$ models is unfortunately a brute-search over the parameter space. However, there is an objective score that a lot of people use to automatically select the optimal parameters.
The formal definition is $$AIC(k)=2k-2ln(\hat{L})$$ where $\hat{L}$ is the maximum likelihood estimate given the model with $k$ parameters.