$R^2$ and adjusted $R^2$ in presence of overlapping observations
$R^2$ and adjusted $R^2$ in presence of overlapping observations
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Richard Hardy · External communityPost link
External question — Cross Validated Stack Exchange
Author: Richard Hardy
Original post: https://stats.stackexchange.com/questions/432173
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
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Given a linear model
$$
y=X\beta+\varepsilon,
$$
the population value of
$R^2$
is
$$
R^2=1-\frac{\text{Var}(\varepsilon)}{\text{Var}(y)}.
$$
The vanilla estimator of
$R^2$
is
$$
\hat R^2=1-\frac{\widehat{\text{Var}}_{biased}(\varepsilon)}{\widehat{\text{Var}}_{biased}(y)}=1-\frac{\frac{1}{n}\sum_{i=1}^n \hat\varepsilon_i^2}{\frac{1}{n}\sum_{i=1}^n (y_i-\bar{y})^2}
$$
and the adjusted estimator of
$R^2$
is
$$
\hat R^2_{adj.}=1-\frac{\widehat{\text{Var}}_{unbiased}(\varepsilon)}{\widehat{\text{Var}}_{unbiased}(y)}=1-\frac{\frac{1}{n-p-1}\sum_{i=1}^n \hat\varepsilon_i^2}{\frac{1}{n-1}\sum_{i=1}^n (y_i-\bar{y})^2}.
$$
This is applicable to the case where the observations of variables do not overlap. Meanwhile, I am interested in the case when they do. Under
overlapping observations
where the overlap is of length
$k$
*, the long-run variance of a generic variable
$x$
(where we may put
$y$
or
$\varepsilon$
in its place as needed) is
$$
\text{LRVar}(x)=\sum_{j=-k}^k \text{Cov}(x_t,x_{t-j})=\text{Var}(x)+2\sum_{j=1}^k \text{Cov}(x_t,x_{t-j})
$$
and some estimators for it (like Newey-West) are available.**
Questions
Should an estimator of the long-run variance be used in estimating
$R^2$
, or should one stick to the regular estimators as in
$\hat R^2$
and
$\hat R^2_{adj.}$
above?
Would the choice of regular variance vs. long-run variance have any effect, given that
$\text{Var}(\varepsilon)$
(or
$\text{LRVar}(\varepsilon)$
) is in the numerator and
$\text{Var}(y)$
(or
$\text{LRVar}(y)$
) is in the denominator, hinting at possible cancellations?
How would the interpretation of these estimators of
$R^2$
(one employing the regular variance estimator and another employing the long-run variance estimator) differ?
*By
overlapping observations
of
$x_t$
where the overlap is of length
$k$
I mean a case where
$x_t=\sum_{\tau=t-k+1}^t \xi_\tau$
where
$\xi_\tau$
is some random process. Hence,
$x_t$
and
$x_{t-\kappa}$
measure partly the same thing for
$\kappa<k$
; they "overlap". An example would be measuring monthly financial returns every day. The monthly return
$x_t$
of today overlaps with the monthly return of yesterday
$x_{t-1}$
to a large degree: given a month with 30 trading days, 29 daily returns
$\xi_{t-29},\dots,\xi_{t-1}$
constitute both
$x_t$
and
$x_{t-1}$
, while only
$\xi_{t}$
and
$\xi_{t-30}$
make
$x_t$
and
$x_{t-1}$
differ. (How many trading days a month has depends on the market.)
**I guess estimating
$\text{LRVar}$
by just plugging in sample counterparts of population quantities may not be a good idea in cases where
$k<<n$
.
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crux26 · External communityPost link
External answer — Cross Validated Stack Exchange
Author: crux26
Original post: https://stats.stackexchange.com/a/432197
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
Adaptation: HTML converted to plain text; contact email addresses removed.
I will refer to population, vanilla, adjusted as (1), (2), (3), respectively.
Q1)
As (1) is for population while (2), (3) are its sample analogue, the same will hold for LRVar. For the population you will use
$k=\infty$
and some integer for the sample.
Q2)
I haven't done the calculation, but using LRVar will make a difference. Given that Newey-West is to take autocorrelation in errors into account, it will have less SE compared to OLS or HC estimators. It returns "more conservative" values, so I presume using LRVar will result in smaller
$R^2$
.
Q3)
Technically they will differ, but in terms of interpretation I wouldn't bother. SEs or p-values does matter, but
$R^2$
are just to denote the overall fit and often the value themselves are not that meaningful. Adjusted
$R^2$
< unadjusted
$R^2$
will hold for both regular variance and long-run variance, so I will just use the regular one, which is easier.
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Quoted from Forex.com.bd-Editorial External question — Cross Validated Stack Exchange Author: Richard Hardy Source score (net votes, not local likes): 6 Original post: https://stats.stackexchange.com/questions/432173 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Given a linear model $$ y=X\beta+\varepsilon, $$ the population value of $R^2$ is $$ R^2=1-\frac{\text{Var}(\varepsilon)}{\text{Var}(y)}. $$ The vanilla estimator of $R^2$ is $$ \hat R^2=1-\frac{\widehat{\text{Var}}_{biased}(\varepsilon)}{\widehat{\text{Var}}_{biased}(y)}=1-\frac{\frac{1}{n}\sum_{i=1}^n \hat\varepsilon_i^2}{\frac{1}{n}\sum_{i=1}^n (y_i-\bar{y})^2} $$ and the adjusted estimator of $R^2$ is $$ \hat R^2_{adj.}=1-\frac{\widehat{\text{Var}}_{unbiased}(\varepsilon)}{\widehat{\text{Var}}_{unbiased}(y)}=1-\frac{\frac{1}{n-p-1}\sum_{i=1}^n \hat\varepsilon_i^2}{\frac{1}{n-1}\sum_{i=1}^n (y_i-\bar{y})^2}. $$ This is applicable to the case where the observations of variables do not overlap. Meanwhile, I am interested in the case when they do. Under overlapping observations where the overlap is of length $k$ *, the long-run variance of a generic variable $x$ (where we may put $y$ or $\varepsilon$ in its place as needed) is $$ \text{LRVar}(x)=\sum_{j=-k}^k \text{Cov}(x_t,x_{t-j})=\text{Var}(x)+2\sum_{j=1}^k \text{Cov}(x_t,x_{t-j}) $$ and some estimators for it (like Newey-West) are available.** Questions Should an estimator of the long-run variance be used in estimating $R^2$ , or should one stick to the regular estimators as in $\hat R^2$ and $\hat R^2_{adj.}$ above? Would the choice of regular variance vs. long-run variance have any effect, given that $\text{Var}(\varepsilon)$ (or $\text{LRVar}(\varepsilon)$ ) is in the numerator and $\text{Var}(y)$ (or $\text{LRVar}(y)$ ) is in the denominator, hinting at possible cancellations? How would the interpretation of these estimators of $R^2$ (one employing the regular variance estimator and another employing the long-run variance estimator) differ? *By overlapping observations of $x_t$ where the overlap is of length $k$ I mean a case where $x_t=\sum_{\tau=t-k+1}^t \xi_\tau$ where $\xi_\tau$ is some random process. Hence, $x_t$ and $x_{t-\kappa}$ measure partly the same thing for $\kappa<k$ ; they "overlap". An example would be measuring monthly financial returns every day. The monthly return $x_t$ of today overlaps with the monthly return of yesterday $x_{t-1}$ to a large degree: given a month with 30 trading days, 29 daily returns $\xi_{t-29},\dots,\xi_{t-1}$ constitute both $x_t$ and $x_{t-1}$ , while only $\xi_{t}$ and $\xi_{t-30}$ make $x_t$ and $x_{t-1}$ differ. (How many trading days a month has depends on the market.) **I guess estimating $\text{LRVar}$ by just plugging in sample counterparts of population quantities may not be a good idea in cases where $k<<n$ .
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