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See also Spring 2004.
The linear regression model is a generalization of our current project.
An application using the data for Stata 3:
The critical t value is a bit larger than the z critical value, but the algebra is identical.
The t table is organized as follows:
We viewed 8.10 as both CI and HT.
From Thursday's class:
The C Section of the Wall Street Journal is largely concerned with how investors are allocating their portfolios among stocks, bonds, and cash. The latter is short-term securities paying interest with very low default risk. 90-day Treasury bills are a good example of cash.
]]> Issues. Your goal is to study five issues.1. Evaluate the risk and return of investing in the stocks, Treasury bonds, and Treasury bills. You should consider both nominal returns and real returns (adjusted for inflation).
2. Evaluate the extent to which returns in these markets are correlated, implying that sophisticated portfolio optimization is possible.
3. Determine how these markets are affected by business cycles as measured by changes in real gross domestic product.
4. Determine how changes in these markets affect physical investment and, hence, business cycles.
5. Assess the extent to which the relationships in 1-4 are stable over time. Identify any interesting subperiods, including a discussion of why particular subperiods might be of interest. Ultimately, this is a very important issue for investors so you should devote half your efforts on this project to it.
Data is available here for the years 1929-2002. Security values are end-of-year. The other numbers are annual aggregates.
tbill - index for the Treasury bill market, including interest payments and changes in security values.
tbond - index for the Treasury bond market, including interest payments and changes in security values.
sp500 - index for the Standard and Poor's 500, including dividend payments and changes in security values.
cpi - the consumer price index.
gdp - gross domestic product.
inv - physical investment.
year - year.
You will want to calculate percentage changes over time. For a variable x, Stata uses l.x to denote the first lag (last year's value). Stata uses d.x to denote x - l.x, or the change since last period, which is also known as the first difference. (The general format is l1.x, l2.x, l3.x for the first, second, and third lags.) The first character in a lag is an "ell". The percentage change in x is thus d.x/l.x.
Do File. Some useful variables are calculated in this do file.
References:
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For #34, the textbook includes a correction for the fact that we are drawing a sample out of a finite population. This is seldom an important concern in empirical work, but it does make a little difference.
Using a 5% Type I error probability, we calculate the rejection region.
The probability of a Type II error depends on the specific value of the unknown parameter under the alternative hypothesis.
Conducting the test with n = 100 is not likely to be very useful in close elections.
Following one of the homework problems, the standard deviation of the sample proportion is 0.0158 for n = 250. We calculated a probability.
The core of the confidence interval concept is the following algebra.
We frequently apply the lemma that p(1-p) = 0.25, more or less.
The true is fixed, but unknown. The confidence interval, which is centered on the sample proportion, moves around (is stochastic). There is a 95% probability that a 95% confidence interval includes the true parameter value.
We considered samples of size 100, 1,000, 10,000, and 1,000,000. At $1 per observation, collecting a sample large enough to accurately estimate p could get expensive.
Please note that the "standard error" of a sample mean
or a sample proportion is a special name applied to a
standard deviation obtained by inference rather than by
direct measurement. That is, the standard error is an
inferred standard deviation. Until this distinction becomes
clear to you, you can simply work on the basis
"standard error = standard deviation".
