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This is largely because the variance of stock returns is high relative to their mean.

The idea that stock return means are harder to estimate is old and was already known before high frequency data, or even GARCH models, were widely used. The point is made e.g. in this 85 paper by Jorion who writes:

On the other hand, uncertainty in variances and covariances is not as critical because they are more precisely estimated

However, I believe the point is even older.

Let me consider a simple example. Assume stock returns are i.i.d. and follow a normal distribution $r \sim N(\mu, \sigma^2)$, where both the mean and variance are unknown. The standard confidence interval for the mean is

$$[\hat{\mu} - t_{n-1,\alpha/2}\frac{s}{\sqrt{n}},\hat{\mu} + t_{n-1,\alpha/2}\frac{s}{\sqrt{n}}],$$

where $t_{n-1,\alpha/2}$ is the $\alpha/2$-percentile t-stat with $n-1$ degrees of freedom. The confidence interval for standard deviation uses the chi-square distribution and is given by (see here)`

$$\left[\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}}},\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}}}\right].$$

Consider the monthly returns of the S&P 500 (long-run mean roughly $0.8\%$ and standard deviation $4.5\%$). Assume you sample 20 years of returns, i.e. $n=240$. Assume your estimators happen to get the mean and standard deviation correct. Now the $95\%$-confidence interval for mean becomes

$$[0.23,1.37].$$

The confidence interval for the standard deviation becomes

$$[4.13,4.94].$$

You can see that the confidence interval for standard deviation is relatively tighter. But this does seem to be the case for arbitrary values of mean and standard deviation. Rather the stock return mean and standard deviation happen to be such that the latter bound is relatively tighter because the mean is low relative to standard deviation.

Note that the confidence interval for standard deviation is not much tighter in absolute terms. If you increase the stock return mean to say $10\%$ monthly holding standard deviation constant, the confidence interval for mean becomes relative tigher than that for the standard deviation. If you look at any other normal distribution, you might easily find that you effectively estimate the mean with greater precision than standard deviation. As the answer by kurtosis suggests, in other contexts, means are often easier to estimate than variances.

This is largely because the variance of stock returns is high relative to their mean.

The idea that stock return means are harder to estimate is old and was already known before high frequency data, or even GARCH models, were widely used. The point is made e.g. in this 85 paper by Jorion who writes:

On the other hand, uncertainty in variances and covariances is not as critical because they are more precisely estimated

However, I believe the point is even older.

Let me consider a simple example. Assume stock returns are i.i.d. and follow a normal distribution $r \sim N(\mu, \sigma^2)$, where both the mean and variance are unknown. The standard confidence interval for the mean is

$$[\hat{\mu} - t_{n-1,\alpha/2}\frac{s}{\sqrt{n}},\hat{\mu} + t_{n-1,\alpha/2}\frac{s}{\sqrt{n}}],$$

where $t_{n-1,\alpha/2}$ is the $\alpha/2$-percentile t-stat with $n-1$ degrees of freedom. The confidence interval for standard deviation uses the chi-square distribution and is given by (see here)`

$$\left[\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}}},\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}}}\right].$$

Consider the monthly returns of the S&P 500 (long-run mean roughly $0.8\%$ and standard deviation $4.5\%$). Assume you sample 20 years of returns, i.e. $n=240$. Assume your estimators happen to get the mean and standard deviation correct. Now the $95\%$-confidence interval for mean becomes

$$[0.23,1.37].$$

The confidence interval for the standard deviation becomes

$$[4.13,4.94].$$

You can see that the confidence interval for standard deviation is relatively tighter. But this is not the case for arbitrary values of mean and standard deviation. Rather the stock return mean and standard deviation happen to be such that the latter bound is relatively tighter because the mean is low relative to standard deviation.

If you increase the stock return mean to say $10\%$ monthly holding standard deviation constant, the confidence interval for mean becomes relative tigher than that for the standard deviation. If you look at any other normal distribution, you might easily find that you estimate the mean with greater precision than standard deviation. As the answer by kurtosis suggests, in other contexts, means are often easier to estimate than variances.

This is largely because the variance of stock returns is high relative to their mean.

The idea that stock return means are harder to estimate is old and was already known before high frequency data, or even GARCH models, were widely used. The point is made e.g. in this 85 paper by Jorion who writes:

On the other hand, uncertainty in variances and covariances is not as critical because they are more precisely estimated

However, I believe the point is even older.

Let me consider a simple example. Assume stock returns are i.i.d. and follow a normal distribution $r \sim N(\mu, \sigma^2)$, where both the mean and variance are unknown. The standard confidence interval for the mean is

$$[\hat{\mu} - t_{n-1,\alpha/2}\frac{s}{\sqrt{n}},\hat{\mu} + t_{n-1,\alpha/2}\frac{s}{\sqrt{n}}],$$

where $t_{n-1,\alpha/2}$ is the $\alpha/2$-percentile t-stat with $n-1$ degrees of freedom. The confidence interval for standard deviation uses the chi-square distribution and is given by (see here)`

$$\left[\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}}},\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}}}\right].$$

Consider the monthly returns of the S&P 500 (long-run mean roughly $0.8\%$ and standard deviation $4.5\%$). Assume you sample 20 years of returns, i.e. $n=240$. Assume your estimators happen to get the mean and standard deviation correct. Now the $95\%$-confidence interval for mean becomes

$$[0.23,1.37].$$

The confidence interval for the standard deviation becomes

$$[4.13,4.94].$$

You can see that the confidence interval for standard deviation is relatively tighter. But this does seem to be the case for arbitrary values of mean and standard deviation. Rather the stock return mean and standard deviation happen to be such that the latter bound is relatively tighter because the mean is low relative to standard deviation.

Note that the confidence interval for standard deviation is not much tighter in absolute terms. If you would increase the stock return mean to say $10\%$ monthly holding standard deviation constant, the confidence interval for mean would become relative tigher than that for the standard deviation. If you look at any other normal distribution, you might easily find that you effectively estimate the mean with greater precision than standard deviation. As the answer by kurtosis suggests, in other contexts, means are often easier to estimate than variances.

This is largely because the variance of stock returns is high relative to their mean.

The idea that stock return means are harder to estimate is old and was already known before high frequency data, or even GARCH models, were widely used. The point is made e.g. in this 85 paper by Jorion who writes:

On the other hand, uncertainty in variances and covariances is not as critical because they are more precisely estimated

However, I believe the point is even older.

Let me consider a simple example. Assume stock returns are i.i.d. and follow a normal distribution $r \sim N(\mu, \sigma^2)$, where both the mean and variance are unknown. The standard confidence interval for the mean is

$$[\hat{\mu} - t_{n-1,\alpha/2}\frac{s}{\sqrt{n}},\hat{\mu} + t_{n-1,\alpha/2}\frac{s}{\sqrt{n}}],$$

where $t_{n-1,\alpha/2}$ is the $\alpha/2$-percentile t-stat with $n-1$ degrees of freedom. The confidence interval for standard deviation uses the chi-square distribution and is given by (see here)`

$$\left[\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}}},\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}}}\right].$$

Consider the monthly returns of the S&P 500 (long-run mean roughly $0.8\%$ and standard deviation $4.5\%$). Assume you sample 20 years of returns, i.e. $n=240$. Assume your estimators happen to get the mean and standard deviation correct. Now the $95\%$-confidence interval for mean becomes

$$[0.23,1.37].$$

The confidence interval for the standard deviation becomes

$$[4.13,4.94].$$

You can see that the confidence interval for standard deviation is relatively tighter. But this does seem to be the case for arbitrary values of mean and standard deviation. Rather the stock return mean and standard deviation happen to be such that the latter bound is relatively tighter because the mean is low relative to standard deviation.

Note that the confidence interval for standard deviation is not much tighter in absolute terms. If you increase the stock return mean to say $10\%$ monthly holding standard deviation constant, the confidence interval for mean becomes relative tigher than that for the standard deviation. If you look at any other normal distribution, you might easily find that you effectively estimate the mean with greater precision than standard deviation. As the answer by kurtosis suggests, in other contexts, means are often easier to estimate than variances.

This is largely due to variance of stock returns being high relative to mean. I think the notion that stock return means are harder to estimate is old and was already known before high frequency data, or even GARCH models, were widely used.

Let me consider a simple example. Assume stock returns are i.i.d. and follow a normal distribution $r \sim N(\mu, \sigma^2)$, where both the mean and variance are unknown. The standard confidence interval for the mean is

$$[\hat{\mu} - t_{n-1,\alpha/2}\frac{s}{\sqrt{n}},\hat{\mu} + t_{n-1,\alpha/2}\frac{s}{\sqrt{n}}],$$

where $t_{n-1,\alpha/2}$ is the $\alpha/2$-percentile t-stat with $n-1$ degrees of freedom. The confidence interval for standard deviation uses the chi-square distribution and is given by (see here)`

$$\left[\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}}},\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}}}\right].$$

Consider the monthly returns of the S&P 500 (long-run mean roughly $0.8\%$ and standard deviation $4.5\%$). Assume you sample 20 years of returns, i.e. $n=240$. Assume your estimators happen to get the mean and standard deviation correct. Now the $95\%$-confidence interval for mean becomes

$$[0.23,1.37].$$

The confidence interval for the standard deviation becomes

$$[4.13,4.94].$$

You can see that the confidence interval for standard deviation is relatively tighter. But this does seem to be the case for arbitrary values of mean and standard deviation. Rather the stock return mean and standard deviation happen to be such that the latter bound is relatively tighter because the mean is low relative to standard deviation.

Note that the confidence interval for standard deviation is not much tighter in absolute terms. If you would increase the stock return mean to say $10\%$ monthly holding standard deviation constant, the confidence interval for mean would become relative tigher than that for the standard deviation. If you look at any other normal distribution, you might easily find that you effectively estimate the mean with greater precision than standard deviation. As the answer by kurtosis suggests, in other contexts, means are often easier to estimate than variances.

This is largely because the variance of stock returns is high relative to their mean.

The idea that stock return means are harder to estimate is old and was already known before high frequency data, or even GARCH models, were widely used. The point is made e.g. in this 85 paper by Jorion who writes:

On the other hand, uncertainty in variances and covariances is not as critical because they are more precisely estimated

However, I believe the point is even older.

Let me consider a simple example. Assume stock returns are i.i.d. and follow a normal distribution $r \sim N(\mu, \sigma^2)$, where both the mean and variance are unknown. The standard confidence interval for the mean is

$$[\hat{\mu} - t_{n-1,\alpha/2}\frac{s}{\sqrt{n}},\hat{\mu} + t_{n-1,\alpha/2}\frac{s}{\sqrt{n}}],$$

where $t_{n-1,\alpha/2}$ is the $\alpha/2$-percentile t-stat with $n-1$ degrees of freedom. The confidence interval for standard deviation uses the chi-square distribution and is given by (see here)`

$$\left[\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}}},\sqrt{\frac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}}}\right].$$

Consider the monthly returns of the S&P 500 (long-run mean roughly $0.8\%$ and standard deviation $4.5\%$). Assume you sample 20 years of returns, i.e. $n=240$. Assume your estimators happen to get the mean and standard deviation correct. Now the $95\%$-confidence interval for mean becomes

$$[0.23,1.37].$$

The confidence interval for the standard deviation becomes

$$[4.13,4.94].$$

You can see that the confidence interval for standard deviation is relatively tighter. But this does seem to be the case for arbitrary values of mean and standard deviation. Rather the stock return mean and standard deviation happen to be such that the latter bound is relatively tighter because the mean is low relative to standard deviation.

Note that the confidence interval for standard deviation is not much tighter in absolute terms. If you would increase the stock return mean to say $10\%$ monthly holding standard deviation constant, the confidence interval for mean would become relative tigher than that for the standard deviation. If you look at any other normal distribution, you might easily find that you effectively estimate the mean with greater precision than standard deviation. As the answer by kurtosis suggests, in other contexts, means are often easier to estimate than variances.