Chapter 13. Bridge to econometrics
Unit 13.2. How to read Dougherty's book
- For each property use a name, like additivity, homogeneity etc. Divide all large proofs into smaller parts that you can understand. Initially it will take a lot of time but will pay itself later. Remember that understanding the first 6 or 7 chapters with proofs is critically important, because in the subsequent chapters most proofs are skipped and you will have to use analogy with what you know from the first several chapters.
- Equation (R.54) and many others: do yourself a favor, use the notation from Unit 10.1 wherever summation is involved, to avoid long, indigestible formulas.
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Equation (R.55): use a two-step definition of a normal variable (first define the standard normal and then a general normal as a linear transformation of the standard normal), as it is done in Unit 6.6.
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Prepare to conquer Dougherty's book in advance by reading my book. Reading the review chapter (that's almost 100 pages) can be avoided if you have mastered my book when taking second year Stats. You should become friends with algebra by doing two-page proofs.
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The relationship between $latex t$ statistic and $latex F$ statistic on p. 147 has a simpler proof, see Comment (2) in Unit 9.6. It is a general probabilistic fact not related to any model.
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Question 14 from Unit 5.18 is related to the logit model from Chapter 10, see equation (10.6) in Dougherty.
Translate all verbal statements to math.
Example 13.4. Reasons to include the disturbance term (Section 1.1 of the book).
Reason 1. It often happens that there are variables that you would like to include in the regression equation but cannot because you are unable to measure them.
Translation. The true model is $latex y_i = \beta_0 + \beta_1 x_i + \beta_2 z_i + u_i$. Omitting $latex z_i$ we get the wrong model $latex y_i = \beta_0 + \beta_1 x_i + v_i$. Looking at it from the point of view of the true model we see that the error becomes worse, $latex v_i = \beta_2 z_i + u_i$. In particular, $latex E v_i = \beta_2 E z_i$ may not be zero, in which case the slope estimator for the wrong model is biased.
Reason 2. In many cases, the relationship is an attempt to summarize in aggregate a number of microeconomic relationships. For example, the aggregate consumption function is an attempt to summarize a set of individual expenditure decisions.
Translation. Aggregation in economics means summation. An individual consumption function models dependence of individual consumption $latex c_{i, t}$ on individual income $latex y_{i, t}$ through
$latex c_{i, t} = \beta_0^i + \beta_1^i y_{i, t} + u_{i, t}. \qquad (\textrm{Eq. 13.1})$
Here the subscript $latex i$ stands for "individual" and $latex t$ stands for "time". ☞ The coefficients $latex \beta_0^i, \beta_1^i$ depend on the individual. In practice, we don't have data on individual consumption and incomes. We have their aggregates $latex C_t = \sum_i c_{i, t}$ and $latex Y_t = \sum_i y_{i, t}$. The equation for aggregate figures obtains from Eq. (13.1) by summing:
$latex C_t = \sum_i c_{i, t} = \sum_i \beta_0^i + \sum_i \beta_1^i y_{i, t} + \sum_i u_{i, t}.$
The intercept $latex \beta_0 = \sum_i \beta_0^i$ for the aggregate equation easily obtains from individual intercepts but the slope $latex \beta_1$ does not. We have to force it into the equation:
$latex \begin{array}{rcl} C_t & = & \beta_0 + \beta_1 \sum_i y_{i, t} + \sum_i (\beta_1^i - \beta_1) y_{i, t} + \sum_i u_{i, t} \\ & = & \beta_0 + \beta_1 Y_t + u_t. \end{array}$
Here the error $latex u_t = \sum_i (\beta_1^i - \beta_1) y_{i, t} + \sum_i u_{i, t}$ includes the errors from microeconomic relationships plus the additional disturbance that arises from aggregation.
Reason 3. The model may be misspecified in terms of its structure.
Reason 4. The functional relationship between $latex Y$ and $latex X$ may be misspecified mathematically.
Translation. Reasons 3 and 4 are better understood if put side by side, which we do in Table 13.1.
Table 13.1. Two types of misspecification.
| Model misspecification | Functional misspecification |
|---|---|
| Wrong variables, right functional dependence | Right variables, wrong functional dependence |
| Example. Inventory $latex I_t$ for year $latex t$ depends on the demand $latex D_t^*$ for that year anticipated (forecasted) by the producer in the previous year. This suggests the true model $latex I_t = \beta_0 + \beta_1 D_t^* + u_t$. However, the forecasts may not be available. By replacing $latex D_t^*$ with the true demand $latex D_t$ for year $latex t$ we get the wrong model $latex I_t = \beta_0 + \beta_1 D_t + v_t$. Obviously, $latex v_t = u_t + \beta_1 (D_t^* - D_t)$. | The true model is nonlinear: $latex \log(y_t) = \alpha \log(x_t) + u_t$ but we attempt to estimate the linear model $latex y_t = \alpha x_t + v_t$. |
Reason 5 (measurement error model). If the measurement of one or more of the variables in the relationship is subject to error, the observed values will not appear to conform to an exact relationship, and the discrepancy contributes to the disturbance term.
Translation. Suppose that the true model is $latex y_i = \beta_0 + \beta_1 x_i + u_i$ and that $latex x_i$ is measured with error. That is, $latex x_i$ is not observed and instead one has $latex x_i^*$ such that $latex x_i = x_i^* + \varepsilon_i$ where $latex \varepsilon_i$ is the measurement error. The true dependence becomes $latex y_i = \beta_0 + \beta_1 x_i^* + (\beta_1 \varepsilon_i + u_i)$. One can regress $latex y_i$ on the observed $latex x_i^*$ but this increases the error.
This discussion shows that one has to read between the lines to understand Dougherty.
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