Serial correlation consistent

In the following commands we are going to simulate two vectors of length , each with a linearly increasing sequence of integers with some normally distributed noise added.

Thus we are constructing linearly associated variables by design. We will firstly construct a scatter plot and then calculate the sample covariance using the cor function. In order to ensure you see exactly the same data as I do, we will set a random seed of 1 and 2 respectively for each variable:.

Scatter plot of two linearly increasing variables with normally distributed noise. There is a relatively clear association between the two variables. We can now calculate the sample covariance:. One drawback of using the covariance to estimate linear association between two random variables is that it is a dimensional measure. That is, it isn't normalised by the spread of the data and thus it is hard to draw comparisons between datasets with large differences in spread.

This motivates another concept, namely correlation. Correlation is a dimensionless measure of how two variables vary together, or "co-vary". In essence, it is the covariance of two random variables normalised by their respective spreads. The following R code will calculate the sample correlation:. The sample correlation is given as 0. Now that we outlined the general definitions of expectation, variance, standard deviation, covariance and correlation we are in a position to discuss how they apply to time series data.

Firstly, we will discuss a concept known as stationarity. This is an extremely important aspect of time series and much of the analysis carried out on financial time series data will concern stationarity. Once we have discussed stationarity we are in a position to talk about serial correlation and construct some correlogram plots. This definition is useful when we are able to generate many realisations of a time series model. However in real life this is usually not the case!

We are "stuck" with only one past history and as such we will often only have access to a single historical time series for a particular asset or situation. So how do we proceed if we wish to estimate the mean, given that we don't have access to these hypothetical realisations from the ensemble?

Well, there are two options:. Now that we've seen how we can discuss expectation values we can use this to flesh out the definition of variance. Once again we make the simplifying assumption that the time series under consideration is stationary in the mean.

With that assumption we can define the variance:. Importantly, you can see how the definition strongly relies on the fact that the time series is stationary in the mean i. You might notice that this definition leads to a tricky situation. If the variance itself varies with time how are we supposed to estimate it from a single time series? Once again, we simplify the situation by making an assumption.

Once we have made this assumption we are in a position to estimate its value using the sample variance definition above:. This is where we need to be careful! With time series we are in a situation where sequential observations may be correlated. This will have the effect of biasing the estimator, i. This will be particularly problematic in time series where we are short on data and thus only have a small number of observations. This is common with time-series data which we will see in the next reading.

This is a serial correlation in which positive regression errors for one observation increases the possibility of observing a positive regression error for another observation. This is serial correlation in which a positive regression error for one observation increases the likelihood of observing a negative regression error for another observation. Autocorrelation does not cause bias in the coefficient estimates of the regression.

However, a positive serial correlation inflates the F-statistic to test for the overall significance of the regression as the mean squared error MSE will tend to underestimate the population error variance. This increases Type I errors the rejection of the null hypothesis when it is actually true. The positive serial correlation makes the ordinary least squares standard errors for the regression coefficients underestimate the true standard errors.

Moreover, it leads to small standard errors of the regression coefficient, making the estimated t-statistics seem to be statistically significant relative to their actual significance.

On the other hand, negative serial correlation overestimates standard errors and understates the F-statistics. This increases Type II errors The acceptance of the null hypothesis when it is actually false. The first step of testing for serial correlation is by plotting the residuals against time. The other most common formal test is the Durbin-Watson test. The Durbin Watson tests the null hypothesis of no serial correlation against the alternative hypothesis of positive or negative serial correlation.

Key Guidelines. Consider a regression output that includes two independent variables that generate a DW statistic of 0.

Assume that the sample size is Adjusting the coefficient standard errors for the regression estimates to take into account serial correlation. This is done using the Hansen method. This method can also be used to correct conditional heteroskedasticity.

Hansen white standard errors are then used for hypothesis testing of the regression coefficient. Consider a regression model with 80 observations and two independent variables. Suppose that the correlation between the error term and a first lagged value of the error term is 0. The most appropriate decision is:. LOS 2 k Explain the types of heteroskedasticity and how heteroskedasticity and serial correlation affect statistical inference. There are two major issues when estimating the required return of equities in Read More.

When evaluating the capital structure of a company, an analyst must consider the Equity Method The equity method of accounting provides a more objective basis for The following figure illustrates homoscedasticity and heteroskedasticity. Types of Heteroskedasticity Unconditional Heteroskedasticity Unconditional heteroskedasticity occurs when the heteroskedasticity is uncorrelated with the values of the independent variables.

Effects of Heteroskedasticity i. It does not affect the consistency of the regression parameter estimators. Use precise geolocation data. Select personalised content. Create a personalised content profile. Measure ad performance. Select basic ads. Create a personalised ads profile. Select personalised ads. Apply market research to generate audience insights. Measure content performance. Develop and improve products. List of Partners vendors. Your Money. Personal Finance.

Your Practice. Popular Courses. Financial Analysis How to Value a Company. What Is a Serial Correlation? Key Takeaways Serial correlation is the relationship between a given variable and a lagged version of itself over various time intervals. It measures the relationship between a variable's current value given its past values. A variable that is serially correlated indicates that it may not be random.

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Related Terms What Is Autocorrelation?