by TheManR
Last Updated February 11, 2019 10:19 AM

I am interested in conducting a time series regression, such that:

$RV_{t+1} = \alpha + (\beta_0 + \beta_1 RET + \beta_2 RV_t)RV_t + \beta_3 RW + \beta_4 RM$.

Is such a regression possible in Python or R? I have seen it in a few papers, and it is apparently estimated through simple OLS, so I feel like I am missing something.

Two papers where this is seen: Conditional Volatility Persistence - Wang and Yang (2018); Exploiting the errors: A simple approach for improved volatility forecasting - Bollerslev, Patton and Quaedvlieg (2015).

This just looks like a way to write interaction terms and polynomials (and is also not specifically related to time series regression). Multiplying out the brackets gives $$RV_{t+1} = \alpha + \beta_0RV_t + \beta_1 RET\cdot RV_t + \beta_2 RV_t^2 + \beta_3 RW + \beta_4 RM $$

Try something like

```
n <- 10
x1 <- rnorm(n)
x2 <- rnorm(n)
y <- rnorm(n)
lm(y~x1*x2+I(x1^2))
```

Output:

```
> lm(y~x1*x2+I(x1^2))
Call:
lm(formula = y ~ x1 * x2 + I(x1^2))
Coefficients:
(Intercept) x1 x2 I(x1^2) x1:x2
0.08859 0.93306 -0.03421 -0.66431 -0.18097
```

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