Short answer is yes, differencing will introduce a negative autocorrelation into the differenced series in most situations. Assuming a mean centered variable to make the notation a bit simpler, the covariance between the differenced series can be represented as:
$$Cov(\Delta X_t,\Delta X_{t-1}) = E[\Delta X_t \cdot \Delta X_{t-1}]$$
Where
- $\Delta X_t = X_t - X_{t-1}$
- $\Delta X_{t-1} = X_{t-1} - X_{t-2}$
Breaking this down into the original variables, we then have:
\begin{align}
E[X_t \cdot X_{t-1}] &= E[(X_t - X_{t-1}) \cdot (X_{t-1} - X_{t-2}) ] \\
&= E[X_tX_{t-1} - X_tX_{t-2} - X_{t-1}X_{t-1} + X_{t-1}X_{t-2}]
\end{align}
The multiplications are then just variances and covariances of the levels:
$$Cov(X_t,X_{t-1}) - Cov(X_t,X_{t-2}) - Var(X_{t-1}) + Cov(X_{t-1},X_{t-2})$$
So here we can see that many different situations will result in negative autocorrelations of the differenced series - basically only in the case that the auto-correlations of the levels are really large (e.g. an integrated series) will the differences have a small negative auto-correlation.
With random data the autocorrelation of the differences will be approximately -0.5, as with random data those covariance terms among the levels will be 0, so it is just $-Var(X_{t-1})$ for the numerator, but with the differences is $Var(X_t) - Var(X_{t-1})$ in the denominator.
This is typically called over-differencing. The solution is to not over-difference the data to begin with.
First of all: Slight Autocorrelation is not unusual for (non-squared) stock return data in my experience. Otherwise there would be no point in trying to decide, whether there is a positive/negative trend or not, from the perspective of investors.
How big is the difference between both $p$-values? Which lag-orders did you choose for the test? The differing values might be from the differing power of the test for the different sample sizes, combined with a weak trend in the in-sample period.
In practice, you may try to include an AR model for the mean series. If the AR coefficients aren't significant (which is to be expected), I'd suggest to just use a fixed mean for the daily return data and model the volatility with some GARCH model that can handle the leverage effect of stock returns (EGARCH, Beta-t-EGARCH).
Best Answer
The big difference is that when the regressors are lags of the dependent variable, the OLS estimator will be inconsistent. In the case of exogenous regressors, the OLS estimator is consistent.
In both cases, the presence of serial correlation will misestimate the standard errors (OLS standard errors will underestimate true standard errors in case of positive serial correlation, and overestimate in case of negative serial correlation) and hence misestimate the t-statistic respectively.
Please note that you test differently for serial correlation depending on whether your regressors are exogenous or not: The standard Durbin-Watson test is perfectly fine when regressors are exogenous but cannot be used with regressors that are lags of the dependent variable.
The question presumes a time series context. Please note that serial correlation occurs in cross-sections as well, therefore your point 1 has a broader context as it may refer to cross sections as well, whereas your point 2 applies only when there are repeated observations over time.
In a time series context, the presence of serial correlation suggests 2 different messages:
Using Hansen or Newey-West serial-correlation robust standard errors only helps with the second point/concern.