Opening the paper…
Linear Statistical Models, Quiz 2
(a)
(b)
(c)
(d)
(e)
Show that is estimable, (where, ) under the linear model if and only if is orthogonal to the Null space of . [4 Marks]
Consider a linear model as
where, .The -th fitted values that result from performing this model is
where
If , then find ? [3 marks]
| 1 | 2 |
| 2 | 3 |
| 3 | 5 |
| 4 | 4 |
| 5 | 6 |
We fit a linear regression model
where .
(a) Find the least square estimates of and . [4 marks]
(b) Calculate the value of RSS (Residual Sum of Squares) [3 Marks]
X = runif(10000,min = 0, max = 20)u = rnorm(10000, mean = 0, sd = 20)y = -2 + 6*X + udf1 = data.frame(X=X, Y=y)l1 = lm(Y~X, data = df1)l1df2 = df[sample(nrow(df1),100),]l2 = lm(Y~X, data = df2)l2
## Output-1Call:lm(formula = Y ~ X, data = df1)
Coefficients:(Intercept) X -1.771 5.981
## Output-2Call:lm(formula = Y ~ X, data = df2)
Coefficients:(Intercept) X 3.278 5.515(a) Explain the functionality of each line of the code. [4 Marks]
(b) Interpret the “Output-1” and “Output-2”. [2 Marks]
(c) Write the fitted linear regression model for both datasets, “df1” and “df2.” Do you observe any differences in the outputs? Provide your comments on the possible reason. [3 Marks]
where, .
Find the value of for which RSS (Residual Sum of Squares) will be minimum. Write all the steps properly. [4 Marks]
1. Suppose $(\underset{\sim}{Y}, X\underset{\sim}{\beta}, \sigma^2 I_{n \times n})$ is a linear model and $p^T\underset{\sim}{\beta}$ is a linear estimable function, where $p \in \mathbb{R}^m$. If there exists a $l \in \mathbb{R}^n$, such that $l^T\underset{\sim}{Y}$ is a linear zero estimator, then which of the following holds? Select all that apply. [2 Marks] (a) $l \in C(X)^\perp$ (b) $p \in \text{columnspace}(X)$ (c) $l \in \text{Null}(X^T)$ (d) $l \in \text{Null}(X)$ (e) $p \in \text{columnspace}(X^T X)$ 2. Show that $\underset{\sim}{p}^T\underset{\sim}{\beta}$ is estimable, (where, $\underset{\sim}{p}^T\underset{\sim}{\beta} = p_1\beta_1 + p_2\beta_2 + \ldots + p_m\beta_m \;\forall\; p \in \mathbb{R}^m$) under the linear model $(\underset{\sim}{Y}, X\underset{\sim}{\beta}, \sigma^2 I)$ if and only if $p$ is orthogonal to the Null space of $X$. [4 Marks] 3. Consider a linear model as $$y_i = \beta_1 x_i + \epsilon_i \quad ; \quad 1 \leq i \leq n$$ where, $\epsilon_i \sim N(0, \sigma^2)$.The $i$-th fitted values that result from performing this model is $$\hat{y_i} = x_i\hat{\beta},$$ where $$\hat{\beta} = \frac{\sum_i^n x_i y_i}{\sum_i^n x_i^2}$$ If $\hat{y_i} = \displaystyle\sum_{j=1}^{n} a_j y_j$, then find $a_j$? [3 marks] 4. Consider the following data points: | $x_i$ | $y_i$ | |---|---| | 1 | 2 | | 2 | 3 | | 3 | 5 | | 4 | 4 | | 5 | 6 | We fit a linear regression model $$y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \qquad i : 1 \to 5,$$ where $\epsilon_i \sim \text{Normal}(0, \sigma^2)$. (a) Find the least square estimates of $\beta_0$ and $\beta_1$. [4 marks] (b) Calculate the value of RSS (Residual Sum of Squares) [3 Marks] 5. Study the code and answer the following sub-questions: X = runif(10000,min = 0, max = 20) u = rnorm(10000, mean = 0, sd = 20) y = -2 + 6*X + u df1 = data.frame(X=X, Y=y) l1 = lm(Y~X, data = df1) l1 df2 = df[sample(nrow(df1),100),] l2 = lm(Y~X, data = df2) l2 ## Output-1 Call: lm(formula = Y ~ X, data = df1) Coefficients: (Intercept) X -1.771 5.981 ## Output-2 Call: lm(formula = Y ~ X, data = df2) Coefficients: (Intercept) X 3.278 5.515 (a) Explain the functionality of each line of the code. [4 Marks] (b) Interpret the “Output-1” and “Output-2”. [2 Marks] (c) Write the fitted linear regression model for both datasets, “df1” and “df2.” Do you observe any differences in the outputs? Provide your comments on the possible reason. [3 Marks] 6. Consider a linear model as $$y_i = \beta_1 x_i + \epsilon_i \quad ; \quad 1 \leq i \leq n$$ where, $\epsilon_i \sim N(0, \sigma^2)$. Find the value of $\beta_1$ for which RSS (Residual Sum of Squares) will be minimum. Write all the steps properly. [4 Marks] 7. A real estate analyst is studying the relationship between the size of a house ($x$, in square feet) and its selling price ( $y$, in thousands of dollars). The analyst collects data from $n$ houses in a city and wants to determine if there is a statistically significant relationship between the size of the house and its selling price. The relationship between the variables is modeled as: $$y_i = \beta_0 + \beta_1 x_i + \epsilon_i \quad ; \quad 1 \leq i \leq n$$ where: - $y_i$ : Selling price of the $i^{th}$ house (in thousands of dollars). - $x_i$ : Size of the $i^{th}$ house (in square feet). - $\beta_0$ : The intercept, representing the average selling price of a house with size $x = 0$. - $\beta_1$ : The slope, representing the change in selling price for every additional square foot of house size. - $\epsilon_i$: Random error terms, assumed to follow a normal distribution $\epsilon_i \sim N(0, \sigma^2)$. The following output has been obtained for a particular dataset : Call: lm(formula = y ~ x, data = house_data) Residuals: Min 1Q Median 3Q Max -6.458 -4.247 -2.036 2.631 12.253 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 26.857367 7.262252 3.698 0.00606 ** x 0.183111 0.002945 62.172 4.98e-12 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 6.654 on 8 degrees of freedom Multiple R-squared: 0.9979, Adjusted R-squared: 0.9977 F-statistic: 3865 on 1 and 8 DF, p-value: 4.98e-12 (a) Define the null and alternative hypotheses. [1 Mark] (b) What are the estimates of $\beta_0$ and $\beta_1$. Interpret the meaning with respect to the given model. [2 Marks] (c) Write the fitted linear regression model. [1 Mark] (d) Based on the output obtained, justify if we can reject the null hypothesis. Also, what can you conclude about the relationship between $x$ (Size of house) and $y$ (Selling price)? [2 Marks]