Derive the ordinary least squares normal equations

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Reference no: EM132578955

Question 1:

The following estimation output was obtained in R (and the table was produced with stargazer). S is the spot exchange rate between sterling and the US dollar, and F is the one-month ahead forward exchange rate between sterling and the US dollar. The monthly data is from bankofengland

OLS estimation, Start = 2001-02-28 End ,= 2019-12-31

Constant

lag(F, -1)

0,016769

(0..020387)
t - 0.822565
p=0.411627
0,959517

(0.012687)
t = 77.992900
p - 0.000000

Observations	227
R²	0.964330
Adjusted R²	0.964172
Residual Std. Error	0.040290 (df= 225)
F Statistic	6,082.892000 (df = 1; 225) (=

The residual sum of squares from the estimation is 0.365239.

Identify the FALSE or INCORRECT statements in the following extract. Provide a brief explanation of your answers.

The sample regression line is

S_t ± AF_t-i±ut

where S_r is the spot exchange rate at time t, and F_t-1 is the one-month ahead forward exchange rate at time t - 1.

There appears to be a negative relationship between St and . The estimated constant is 0.016769.

To test the hypothesis that the slope coefficient is zero, we can use a t test. It is a two-variable model estimated using monthly data, so the appropriate degrees of freedom for this test are 12 - 2 = 10. The calculated t statistic of 77.9929 and p-value of 0.000000 indicate we would reject the hypothesis that the slope coefficient is equal to zero at the 0.05 significance level. A 95% confidence interval for the slope coefficient would include zero. The model does not explain 96.43% of the variation in St. The total sum of squares is less than 0.365239. Multicollinearity is not a problem with this specification.

Question 2:

Consider the two-variable regression model for which the population regression equation can be written in the following form:

Y_i=β₁+β₂x_i+u_i (1)

where Viand X_i are observable variables, fl₁ and fl₇are unknown regression coefficients, and u, is an unobservable population disturbance term.

The corresponding ordinary least squares sample regression equation is given by:

Y_i = β₁+ β₂X_i,+e_i (2)

where β₁ and β₂, are the estimators of the constant and the slope parameter respectively, and ei is the sample residual for the ith observation.

a. Specify the ordinary least squares estimation criterion.
b. Derive the ordinary least squares normal equations to estimate equation (1). Explain your answer.
c. Discuss the conditions under which the estimators fi and )82 obtained from the equations derived in 2(b) are BLUE.

Question 3:

The tab-delimited text file M430 _A S4_2020_Q3.txt contains weekly data on the stock price of the Union Pacific Corporation (UNP), the parent company of Union Pacific Railroad, and the Standard & Poor's 500 index (SP500) from 1 February 2016 to 27 January 2020, giving 209 observations (the share price and stock index are closing prices, in US dollars, source: https://finance.yahoo.com). For reference, for observation 1/2/2016 UNP = 75.03 and SP500 = 1880.05. For 8/2/2016 UNP = 77.20 and SP500 = 1864.78. And for 27/1/2020 UNP = 179.42 and SP500 = 3225.52. Use a 5% significance level in your analysis.

a. Obtain a scatter plot of the weekly log return on Union Pacific stock

(lnUNP_t -In UNP_t-1 )

against the weekly log return on the S&P 500 index for the complete sample, and comment on the plot.

b. Use R to estimate the following model using ordinary least squares and all available observations, and submit your output as your answer:

In (lnUNP_t - ln UNP_t-1 ) = β₁ + β₂ (ln SP500_t - lnSP500_t-1) + μ_t

cl(log(UNP))

0_001528
(0_001577)

= 0.968927 p = 0333718

1_026523
(0_092264)
t= 11_12.5950
p = 0_000000

Constant

d(log(SP500))

Observation 208

R² 0_375354

Adjusted R² 03 72322

Residual Std. Error 0_022473 (df= 206)

F Statistic 12.1786900 (df= 1; 206) (p = 0_000000)

c. Comment on the estimated parameters, including the relationship between the variables and the size of the estimated coefficient β₂

d. (i) Test whether the parameter β₂ is significantly different from zero.
(ii) Test whether the parameter β₂ is greater than 1.

e. Construct and interpret a 95% confidence interval for the parameter β₁.

f. Explain what the value of R² means in the context of the estimation of this model. Use the F test to test whether the calculated R² you obtain in part b) is significantly different from zero.

Reference no: EM132578955

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