What is meant by statistical inference what is its function

Question

Part -1:

1.

(a) What is meant by statistical inference? What is its function and importance?

(b) What is meant by and what is the relationship between a parameter and a statistic?

(c) What is meant by estimation?

(d) What is meant by Hypothesis testing?

2. What is meant by (a) A point estimate? (b) Unbiased estimator? (c) An interval estimate?

3. Given the data below on Consumption and Income, estimate the values of β₀ and β₁ .Clearly show your work.

Y(Consumption)	3172	3124	1951	3269
X(Income)	159	158	42	58

4. A producer of steel cables wants to test if the steel cables it produces have a breaking strength of 5000 lb. A breaking strength of less than 5000 1b would not be adequate, and to produce steel cables with breaking strengths of more than 5000 1b would unnecessarily increase production costs. The producer takes a random sample of 64 pieces and finds that the average breaking strength is 5100 1b and the sample standard deviation is 4801b. Should the producer accept the hypothesis that its steel cable has a breaking strength of 5000 lb at the 5% level of significance?

5. The variable smokes is a binary variable equal to one if a person smokes, and zero otherwise. Using the data in SMOKE.RAW, we estimate a linear probability model for smokes:

Smokes=0.656-0.069 log (cigpric)+ 0.012 log (income)-0.029educ + 0.020age -0.101 restaurn- 0.026white
N=800 R-squared=0.65

The variable white equals one if the respondent is white, and zero otherwise.

(a) Holding other factors fixed, if education increases by four years, what happens to the estimated probability of smoking?

(b) Interpret the coefficient on the binary variable restaurn (a dummy variable equal to one if the person lives in a state with restaurant smoking restrictions).

(c) Interpret R-squared according to the information given above.

6. The following two equations represent a simple wage-price model:

W_t = α₀ + α₁P_t + α₂Q_t + μ_1t

P_t = b₀ + b₁W_t + μ_2t

Where W_t is the wage in time period t , P represents prices, and Q is productivity.

(a) Why is this a simultaneous -equations model?

(b) Which are the endogenous and exogenous variables in the model above?

(c) Why would the estimation of W and P equations by OLS give biased and inconsistent parameter estimates?

7. Prove the following;

(a) Variance(u_i / X_i ) = σ²

(b) Cov(u_i, u_j/X_i, X_j) = 0

(c) Var(X) = Σ(x-x')² /n-1

(d) E(βo ) =  βo

Part -2:

1. What is meant the following terms below?

a) Zero Covariance
b) Standard Error
c) Correlation Analysis
d) White Noise

2. In a recent article, Evans and Schwab (1995) studied the effects of attending a Catholic high school on the probability of attending college. For concreteness, let college be a binary variable equal to unity if a student attends college, and zero otherwise. Let CathHS be a binary variable equal to one if the student attends a Catholic high school. A linear probability model is

College = β₀ + β₁CathHS + Other factor + μ

where the other factors include gender, race, family income, and parental education.

i. Why might CathHS be correlated with u?

ii. Evans and Schwab have data on a standardized test score taken when each student was a sophomore. What can be done with these variables to improve the ceteris paribus estimate of attending a Catholic high school?

iii. Let CathRel be a binary variable equal to one if the student is Catholic. Discuss the two requirements needed for this to be a valid IV for CathHS in the preceding equation. Which of these can be tested?

iv. Not surprisingly, being Catholic has a significant effect on attending a Catholic high school. Do you think CathRel is a convincing instrument for CathHS?

3. Consider the following model:

Y= β₀ + β₁ Education + β₂ year of experience + μ

Suppose you leave out the years of experience variable.

a) What kinds of problems or biases would you expect? Explain thoroughly.

b) Discuss how the Likelihood ratio, Wald, and Lagrange Multiplier (score) tests are different and/or similar.

4. What do the following presentations mean?
(i) ARIMA(0,2,1)
(ii) ARIMA(1,0,0)
(iii) ARMA(1,1)
(iv) MA(1)

QUESTION 5

a) In econometrics, ordinary least squares (OLS) is regarded as a standard and is recommended when all the CLRM assumptions hold. One such assumption is that the explanatory variables be fixed in repeated samples. List three possible causes of the violation of this assumption

b) Explain the difference between an autoregressive and a moving-average process. Why are AR and MA processes referred to as stationary processes?

c) What are the problems of a non-stationary series? What types of variables are likely to be non-stationary?

Answer 1

For the following set of questions, provide responses to each part by applying the proper concepts and theories. Question 1: a) What do you meant by the statistical inference? Describe its function and significance? b) What do you meant by and determines the relationship between the parameter and a statistic? c) What do you meant by the term estimation? d) What do you meant by the Hypothesis testing? Question 2: What do you meant by (i) A point estimate? (ii) Unbiased estimator? (iii) An interval estimate?