Math Problems Vault

Problem 10.1

The linear regression model has the following special form:

X_{k} = \theta _{1}+h_{k} \theta _{2}+\varepsilon _{k}, \quad k = 1, \ldots , n,

where $\left\{ \varepsilon_{k}\right\}$ are independent centered random variables. Find an explicit analytical expression for the OLS estimate $\widehat{\theta }_{n}$ of the vector $\theta = \left\{ \theta_{1}, \theta_{2}\right\}^{\top }$ .

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Problem 10.2

Prove relation (10.7):

\sum _{k = 1}^{n}\left(X_{k}-\bar{X}_{n}\right)^{2} = \sum _{k = 1}^{n}\left(\widehat{X}_{n}-\bar{X}_{n}\right)^{2}+\sum _{k = 1}^{n} \widehat{\varepsilon }_{k}^{2}

where $\bar{X}_{n} = \sum_{k = 1}^{n} X_{k}, \widehat{X}_{k} = h_{k}^{\top } \widehat{\theta }_{n}, \widehat{\varepsilon }_{k} = X_{k}-\widehat{X}_{k}$ .

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Problem 10.3

Let $X = H \theta +E$ , where $\mathbb {E}\left[E\right] = 0, \operatorname {Cov}\left[ E, E \right] = \sigma^{2} I, H \in \mathbb {R}^{n \times p}$ . Prove that if $H$ and $\theta$ are partitioned into blocks in the form

X \theta =\left[\begin{smallmatrix} H_{1} & H_{2} \end{smallmatrix}\right]\left[\begin{smallmatrix} \theta _{1} \\ \theta _{2} \end{smallmatrix}\right]

then the OLS estimate of $\theta_{2}$ has the form: $\widehat{\theta }_{2} = \left[H_{2}^{\top } H_{2}-H_{2}^{\top } H_{1}\left(H_{1}^{\top } H_{1}\right)^{-1} H_{1}^{\top } H_{2}\right]^{-1}\left[H_{2}^{\top } X-H_{2}^{\top } H_{1}\left(H_{1}^{\top } H_{1}\right)^{-1} H_{1}^{\top } X\right]$ .

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Problem 10.4

From the two observations $X_{1} = 2, X_{2} = 4$ with $h_{1} = 1, h_{2} = 2$ , construct a realization of the OLS estimate of the unknown parameters in the simple linear regression model. Compute the coefficient of determination.

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Problem 10.5

In the setting of Example 10.4, use Fisher's test to test the hypothesis $H_{0}: \theta_{1} = -1, \theta_{2} = 2$ at the significance level $\alpha = 0,01$ , assuming that the observation errors have the distribution $\mathcal{N}\left(0 ; \sigma^{2}\right)$ .

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Problem 10.6

The observation model has the form

X_{k} = k \theta +\varepsilon _{k}, \quad k = 1, \ldots , n,

where $\left\{ \varepsilon_{k}\right\}$ are independent Gaussian random variables, $\varepsilon_{k} \sim \mathcal{N}\left(0 ; \sigma^{2}\right), \sigma > 0$ . Using the OLS estimate $\widehat{\theta }_{n}$ of the parameter $\theta$ , construct a confidence interval for it with confidence level $q = 0,95$ .

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Problem 10.7

In the setting of Example 10.4, test at the significance level $p = 0,05$ the parametric hypothesis $H_{0}: \theta_{2} = 0$ , assuming that the observation errors are independent and have the distribution $\mathcal{N}\left(0 ; \sigma^{2}\right)$ .

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Problem 10.8

(1)(2)

Using the data of Example 10.6, construct the dependence $X = \theta_{1}+\theta_{2} h$ , assuming that:

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(1)

$h = [1, \ldots , 12]^{\top }$ ;

(2)

$h = [1954, \ldots , 1967]^{\top }$ .

In both cases, estimate the error of the obtained realization of the estimate and compute the coefficient of determination.

Problem 10.9

Estimate the unknown parameters $\alpha , \beta$ in the model of a person's consumption of some product $v = \alpha +\beta \ln w$ , where $v$ is the expenditure on the consumer product, $w$ is the weekly income (Table 10.4).

Table 10.4

$k$	1	2	3	4	5	6	7
$w_{k}$	0,8	1,2	1,5	2,2	2,3	3,6	3,1
$v_{k}$	1,7	2,7	3,6	5,7	6,7	8,1	12,0

Estimate the elasticity coefficient $\sigma = \frac{d v}{d w} \cdot \frac{w}{v}$ . Assuming that the observation errors have the distribution $\mathcal{N}\left(0 ; \sigma^{2}\right)$ , use Fisher's test to test the hypothesis $H_{0}: \beta = 1$ at the significance level 0,05.

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