Consider an observed time series $\{Y_t\}_{t=1}^T$ and averaged values $$ Y_t^{(h)}=\frac{1}{h} \sum_{i=0}^{h-1} Y_{t-i} $$ and what is called an HAR model (this is a specific example) $$ Y_{t+1}=c+\beta_1 Y_t^{(h_1)}+\beta_2 Y_t^{(h_2)}+\beta_3 Y_t^{(h_3)}+\varepsilon_{t+1} $$ this can be seen as a restricted $AR(p)$ model where $p=\max\{h_1,h_2,h_3\}$. It is claimed to me that this type of model can be estimated by ordinary least squares. Which assumptions do I need to impose for this to be a consistent estimation? Simply the usual zero mean error and error uncorrelated with explanatory? Maybe if someone could give a reference for why it works for the $AR$ and why doing this averaging thing does not mess anything up it would be great!



To consider a concrete example, set $h_1 = 1, h_2 = 2, h_3 = 3$. Then

$$Y_{t+1}=c+\beta_1 Y_t^{(h_1)}+\beta_2 Y_t^{(h_2)}+\beta_3 Y_t^{(h_3)}+\varepsilon_{t+1}$$ becomes

$$Y_{t+1}=c+\beta_1 \frac{1}{h_1} \sum_{i=0}^{h_1-1} Y_{t-i}+\beta_2 \frac{1}{h_2} \sum_{i=0}^{h_2-1} Y_{t-i}+\beta_3 \frac{1}{h_3} \sum_{i=0}^{h_3-1} Y_{t-i}+\varepsilon_{t+1}$$

$$=c+\beta_1 Y_{t}+\frac{\beta_2 }{2}\left(Y_{t}+Y_{t-1} \right)+ \frac{\beta_3}{3} \left(Y_{t}+Y_{t-1}+Y_{t-2}\right)+\varepsilon_{t+1}$$

$$= c+ \left(\beta_1+\frac{\beta_2 }{2}+\frac{\beta_3}{3}\right)Y_t + \left(\frac{\beta_2 }{2}+\frac{\beta_3}{3}\right)Y_{t-1}+\frac{\beta_3}{3}Y_{t-2}+\varepsilon_{t+1}$$

$$=c+ \gamma_1Y_t + \gamma_2Y_{t-1}+\gamma_3Y_{t-2}+\varepsilon_{t+1}$$

This is a standard $AR(3)$ model with drift. Estimation will give you the three gamma-coefficients from which you can derive a unique estimate for the beta-coefficients.

The assumption that the error term is white noise (zero-mean, serially uncorrelated), will make the regressors uncorrelated with future error terms, and the OLS estimator will be consistent.


Such a model may be estimated by ordinary least squares, in similar fashion to an ordinary AR, which may also be estimated by OLS (see the second paragraph here); it works for this model works for exactly the same reasons.

In this case, you construct response and predictors as follows:

$Y = \{Y_{p+1},Y_{p+2},\ldots,Y_T\}$

$X_1 = \{Y^{h_1}_{p},Y^{h_1}_{p+1},\ldots,Y^{h_1}_{T-1}\}$
$X_2 = \{Y^{h_2}_{p},Y^{h_2}_{p+1},\ldots,Y^{h_2}_{T-1}\}$
$X_3 = \{Y^{h_3}_{p},Y^{h_3}_{p+1},\ldots,Y^{h_3}_{T-1}\}$

And then regress $Y$ on $(X_1,X_2,X_3)$

This conditions on the first $p$ $Y$-values, just as with OLS estimation of an $AR(p)$.

A useful way to look at it is to decompose the likelihood for $Y_{1:T}$ as $f(Y_{T}|Y_{1:T-1})\cdot f(Y_{T-1}|Y_{1:T-2})\cdot f(Y_{T-2}|Y_{1:T-3}) \cdot \ldots \cdot f(Y_{p+1}|Y_{1:p})\cdot f(Y_{1:p})$.

OLS estimation corresponds to optimizing an approximation to the likelihood based on all but the final term. As $n\to\infty$, the relative impact on the likelihood of the first $p$ terms becomes less and important - the likelihood is dominated by the last $n-p$ terms (one can formalize this line of argument somewhat).


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