Statistical Ts & Ps
来源：Kaplan Schweser 作者：Dr. Greg Filbeck 时间：2008-09-09 点击：
This week on the online program we covered approximately half of the Quantitative Analysis section of the FRM exam. Quantitative Analysis will make up 10 percent (14 questions) of your exam. To stay on the suggested study schedule, you should plan to finish studying this material over the next week. Quantitative Analysis consists of three broad topic areas: quantitative fundamentals (probability, statistics, and regression analysis); statistical properties and forecasting of correlation, covariance, and volatility; and Monte Carlo simulation and Extreme Value Theory.
In this week's blog posting, we will focus on statistical testing.
Candidates often get T's and P's confused when conducting statistical testing. I think the specific confusion results from lacking a plan of attack when conducting statistical testing.
To keep things simple when studying statistical testing, I suggest a three-prong approach:
1) State the test,
2) Perform the test, and
3) Conclude the test. Let's take a look at each prong sequentially.
1) State the test:
Stating the test involves formally stating a hypothesis. The specific hypothesis you state is totally up to you, and consists of two parts, a null and alternative. You can think of the hypothesis test as evaluating the situation in which a certain relationship exists. For example, does the mean equal 5? Hypothesis tests allow you to address questions such as is there a statistical relationship between housing starts and interest rates?
You can think of the null hypothesis as the situation you are looking to reject, e.g. no relationship between housing starts and interest rates. If you can reject the null hypothesis, you do so in favor of the alternative, e.g. there exists a relationship between housing starts and interest rates. If you cannot reject the null hypothesis given the data you are analyzing, you fail to reject the null. That does not mean you "accept" the null, you simply fail to reject it as a hypothesis regarding the relationship you are analyzing.
When you reject a null in favor of the alternative, but the null is the true relationship, you commit a Type 1 error. When you fail to reject the null, and it does not represent the true relationship stated, you commit a Type 2 error. We need to focus on one or the other error types. Most efforts focus on Type 1 errors, which you should expect being the type of errors addressed on the exam.
2) Perform the test:
This step is the one in which most candidates get bogged down.
The easiest way to think of this step is to evaluate how "far away" the calculated value for your variable of interest is away from the hypothesized value. This "far away" distance could also be in proportional terms, e.g. when comparing two variances. For the moment, let's focus on how one might perform a test indicating whether mean stock returns are statistical different from zero. The null hypothesis would be that mean stock returns are equal to zero, and the alternative would be that they are not equal to zero.
To see how "far away" our average stock returns are away from zero we would subtract the calculated mean from our hypothesized value, e.g. 0.02 - 0, for instance. You might be tempted to say, "Well, the calculated value is not equal to 0, hence, stock returns are not equal to zero." You would be mistaken, however, because to get an idea how "far away" 0.02 is from zero, we also have to take into account the variability of the data. This is done by dividing the difference between the calculated value and the hypothesized value by the data's standard error, which is usually the sample's standard deviation divided by the number of observations analyzed. For simplicity sake, use 0.009 as the standard error in this example.
Once you divide the difference by the standard error, you get your "Test" statistic. That's all there is to performing the test. Calculate a test statistic, and label it "T," which is not to be confused with the t-distribution's "t." T is nothing other than the difference between our calculated value (0.02) and the null hypothesis value (0), divided by the sample's standard error (0.009), which in this case equals 2.22. When you perform the test, think "T = 2.22," nothing more.
3) Conclude the test:
Now that you have a test statistic value (T) of 2.2, what do you do next?
Well, the easiest thing to do is to compare T against a reference value so you can make a reject or fail to reject decision regarding the hypothesis statement. That reference value is often called the "critical value" given by a statistical table. That critical value is connected to the amount of times you are willing to commit a Type 1 error, e.g. 5%. Since statements are often made in "confidence" terms, you may see "test at 95% confidence" on the exam. Both references are saying the same thing, so do not let that fluster you. The critical value from a Standard Normal Z-table relevant for a 95% confidence, or a 5% error is 1.96.
Since the calculated T is 2.2, and the critical value is 1.96, one can reject the null hypothesis that mean stock returns are zero in favor of the alternative that mean returns are not equal to zero. Notice, you are not making a statement regarding the direction of calculated returns, only that they are not equal to zero, hence the reason you usually see "calculate the absolute value of the test statistic before comparing to the critical value."
If your calculated T equals 1.96, you would reject the null in favor of the alternative knowing that you expect to be wrong 5 out of 100 times you do the calculation. Since T is greater than the critical value, you can assume this particular analysis will cause you to commit Type 1 errors less than 5% of the time. Specifically, the exact proportion of Type 1 errors you would make is represented by the "p-value" often generated by statistical packages. For argument sake, assume the p-value for this particular test is presented as 0.015, which is less than 0.05. This p-value means you expect to commit a Type 1 error 1.5 times out of 100.
So, your conclusion with respect to this example is that you reject the null hypothesis in favor of the alternative.
In summary, when conducting statistical tests, do not forget your T's and P's. The rule you can follow when conducting statistical tests is that if your T (test) statistic is greater than the critical value (from statistical tables), you can reject the null in favor of the alternative. On the other hand, if the "p-value" (generated by statistical packages) is less than the level of significance, you can also reject the null in favor of the alternative.
In this week's blog posting, we will focus on statistical testing.
Candidates often get T's and P's confused when conducting statistical testing. I think the specific confusion results from lacking a plan of attack when conducting statistical testing.
To keep things simple when studying statistical testing, I suggest a three-prong approach:
1) State the test,
2) Perform the test, and
3) Conclude the test. Let's take a look at each prong sequentially.
1) State the test:
Stating the test involves formally stating a hypothesis. The specific hypothesis you state is totally up to you, and consists of two parts, a null and alternative. You can think of the hypothesis test as evaluating the situation in which a certain relationship exists. For example, does the mean equal 5? Hypothesis tests allow you to address questions such as is there a statistical relationship between housing starts and interest rates?
You can think of the null hypothesis as the situation you are looking to reject, e.g. no relationship between housing starts and interest rates. If you can reject the null hypothesis, you do so in favor of the alternative, e.g. there exists a relationship between housing starts and interest rates. If you cannot reject the null hypothesis given the data you are analyzing, you fail to reject the null. That does not mean you "accept" the null, you simply fail to reject it as a hypothesis regarding the relationship you are analyzing.
When you reject a null in favor of the alternative, but the null is the true relationship, you commit a Type 1 error. When you fail to reject the null, and it does not represent the true relationship stated, you commit a Type 2 error. We need to focus on one or the other error types. Most efforts focus on Type 1 errors, which you should expect being the type of errors addressed on the exam.
2) Perform the test:
This step is the one in which most candidates get bogged down.
The easiest way to think of this step is to evaluate how "far away" the calculated value for your variable of interest is away from the hypothesized value. This "far away" distance could also be in proportional terms, e.g. when comparing two variances. For the moment, let's focus on how one might perform a test indicating whether mean stock returns are statistical different from zero. The null hypothesis would be that mean stock returns are equal to zero, and the alternative would be that they are not equal to zero.
To see how "far away" our average stock returns are away from zero we would subtract the calculated mean from our hypothesized value, e.g. 0.02 - 0, for instance. You might be tempted to say, "Well, the calculated value is not equal to 0, hence, stock returns are not equal to zero." You would be mistaken, however, because to get an idea how "far away" 0.02 is from zero, we also have to take into account the variability of the data. This is done by dividing the difference between the calculated value and the hypothesized value by the data's standard error, which is usually the sample's standard deviation divided by the number of observations analyzed. For simplicity sake, use 0.009 as the standard error in this example.
Once you divide the difference by the standard error, you get your "Test" statistic. That's all there is to performing the test. Calculate a test statistic, and label it "T," which is not to be confused with the t-distribution's "t." T is nothing other than the difference between our calculated value (0.02) and the null hypothesis value (0), divided by the sample's standard error (0.009), which in this case equals 2.22. When you perform the test, think "T = 2.22," nothing more.
3) Conclude the test:
Now that you have a test statistic value (T) of 2.2, what do you do next?
Well, the easiest thing to do is to compare T against a reference value so you can make a reject or fail to reject decision regarding the hypothesis statement. That reference value is often called the "critical value" given by a statistical table. That critical value is connected to the amount of times you are willing to commit a Type 1 error, e.g. 5%. Since statements are often made in "confidence" terms, you may see "test at 95% confidence" on the exam. Both references are saying the same thing, so do not let that fluster you. The critical value from a Standard Normal Z-table relevant for a 95% confidence, or a 5% error is 1.96.
Since the calculated T is 2.2, and the critical value is 1.96, one can reject the null hypothesis that mean stock returns are zero in favor of the alternative that mean returns are not equal to zero. Notice, you are not making a statement regarding the direction of calculated returns, only that they are not equal to zero, hence the reason you usually see "calculate the absolute value of the test statistic before comparing to the critical value."
If your calculated T equals 1.96, you would reject the null in favor of the alternative knowing that you expect to be wrong 5 out of 100 times you do the calculation. Since T is greater than the critical value, you can assume this particular analysis will cause you to commit Type 1 errors less than 5% of the time. Specifically, the exact proportion of Type 1 errors you would make is represented by the "p-value" often generated by statistical packages. For argument sake, assume the p-value for this particular test is presented as 0.015, which is less than 0.05. This p-value means you expect to commit a Type 1 error 1.5 times out of 100.
So, your conclusion with respect to this example is that you reject the null hypothesis in favor of the alternative.
In summary, when conducting statistical tests, do not forget your T's and P's. The rule you can follow when conducting statistical tests is that if your T (test) statistic is greater than the critical value (from statistical tables), you can reject the null in favor of the alternative. On the other hand, if the "p-value" (generated by statistical packages) is less than the level of significance, you can also reject the null in favor of the alternative.