Rachel: The autocorrelations are zero; these are the theoretical values. The sample autocorrelations are the observed values; because of random fluctuations, they are not exactly zero. Instead, they are normally distributed with a mean of zero and a standard deviation of 1//T.

Rachel: Suppose a time series has 400 values, and we observe sample autocorrelations for the first 100 lags. We have 100 numbers for a normal distribution with a mean of 0 and a standard deviation of 1//T = 1/20. Using the cumulative normal distribution table, we infer that 5% of the autocorrelations should have absolute values of 1.96 × (1/20) . 10% or greater. If the actual percentage of 6% or 4%, we presume the time series is a white noise process. If the actual percentage if 15%, we presume the time series is not a white noise process.

according to last question:

if the actual percentage >=10%,we presume the time series is not white noise process;

if the actual percentage <10%,we presume the time series is a white noise process. is this right? why?

how to determine if a time series is a white noise process?

thanks in advance.

Jacob: How do we work the Bartlett’s problems? Suppose we have 225 observations and a significance level of 10%.

Rachel: Bartlett’s test uses 2 sided Z values.

We use 2 sided because the sample autocorrelation may be positive or negative.

We use Z values, not t values, since we know the variance if the time series is white noise.

The Z value for a significance level of 10% is 1.645. The standard deviation for a time series with 225 observations is 1 / /225 = 1/15 = 6.67%. The sample autocorrelation at a 90% confidence interval is 1.645 × 6.67% = 10.97% . 11%.

Jacob: Does that mean that if the sample autocorrelation is more than 11% the time series is not white noise?

Rachel: We can make only statistical statements: If the time series is white noise, the probability is less than 10% of getting a sample autocorrelation whose absolute value is more than 11%.

Jacob: Why does the textbook use 5% for the regression analysis course and 10% for the time series course?

Rachel: The textbook says this is the common practice. One rationale is that we can always add observations in most regression analysis projects. If we do a social science study and get a p-value of 8.5%, we say: "Let’s do some more interviews or collect more data to see if we can get a p-value less than 5%."

For time series, we can’t add observations. If we have a time series of 100 quarters, we must wait another five years to get 120 quarters. Many p-values are between 5% and 10%, so we use 10%.

A second rationale is that many regression analysis studies seek true causes. To know what increases sales, we may examine several dozen independent variables. To know the true cause, we use a strong significance level.

Most time series studies are proxies: we don’t know the true causes, but we try to find a simple formula. We do not need to be sure of the time series because it is just a proxy.

A third rationale is that most regression analysis studies are theory. A study may examine why citizens vote Democratic or Republican, looking at a dozen independent variables. The study is aimed at policy wonks and academics, who want to know the true relation. If we are not sure, we just say: "I don’t know."

Most time series studies are aimed at business persons, who want to know whether to raise or lower prices or raise or lower advertising. There is no middle ground. A business person must give a recommendation; one can’t say "I don’t know." We use a less stringent significance level.