# Bucharest University Economics Regression Equations Question

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Number II
I (3p). Let the following be a simultaneous equation model that present the demand and supply for a product based on
its price, the income and wealth of the consumers:
(D )
– demand function: Qt = ? 0 + ? 1 Pt + ? 2 I t + ? 3Wt + u1t
(S )
– supply function: Qt
= ? 0 + ?1 Pt + u 2t
(D )
= Qt( S )
– equilibrium relation: Qt
Requirements:
a) Present the types of variables and types of equalities in the model; (0,5p)
b) Write the model in the reduced form; (1p)
c) Identify each regression equation based on the order condition; (1p)
d) For each exactly- or over-identified equation describe the optimal method (methods) for estimating its
parameters. (0,5p)
II (2p).
a) Let
Yt be a time series and lets assume the time series is generated by the following stochastic process:
Yt = ?Yt ?1 + u t . If the value of the estimator of ? is 0.7, and the variance of this estimator is 0.09, decide
based on the Dickey-Fuller test whether the stochastic process is stationary or non-stationary. The critical
value for applying the test for a significance level of 5% is -1,32; (1p)
b) Let
u? t be the residual variable values obtained from a simple regression model:
u? t : 1 -0.8
0.5
1
0.9
-0.8
-1
-1
0.6
-0.4
Verify the hypothesis of non-autocorrelation for the residual variable based on a Durbin-Watson test (
d1 = 0.93 and d 2 = 1.33 for a significance level ? = 5% ). (1p)
III (4p). We know that the returns on equities depend on the returm on the market. In this context, we present in the
following Table the return on equity, Y, for a fictional company and the return to the market, X, for 5 consecutive
months:
Table 1.
Luna
Return on equity for the fictional
company (%)
1
2
3
4
5
Return on the market (%)
5
3
6
6
5
8
5
9
7
6
For your convenience, here are some summary statistics:
,
,
,
,
.
Requirements:
a) Estimate and interpret the parameters
b)
? 0 and ? 1 (estimate using the OLS method); (0,5p)
Determine the confidence intervals for ? 0 and ? 1 for a confidence level of 95% ( t 0.025; 3 = 3.182 ); (1p)
c)
Test the hypothesis that the true parameter
? 1 is equal to 2 ( t 0.025; 3 = 3.182 ); (0,25p)
(
)
d) Test the validity of the model using the Fisher test F0.05;1;3 = 10.13 ; (0,75p)
e)
f)
( )
2
Calculate and interpret the coefficient of determination R ; (0,5p)
Predict the next month return on equity, if the market return will be 10%. Use interval estimators and predict both
the expected return on equity and the individual return on equity (the significance level considered is 5%). (1p)
Number I
I (3p). Let the following be a simultaneous equation model:
Ct = ? 0 + ? 1 I t + u1t
S t = ? 0 + ?1 I t + ? 2 I t ?1 + u 2t
I t = Ct + S t
where C t represents private consumption, I t – income, I t ?1 – income in the previous period (it is assumed
predetermined), S t represents savings.
Requirements:
a) Present the types of variables and types of equalities in the model; (0,5p)
b) Write the model in the reduced form; (1p)
c) Identify each regression equation based on the order condition; (1p)
d) For each exactly- or over-identified equation describe the optimal method (methods) for estimating its
parameters. (0,5p)
II (2p).
a) Let
Yt be a time series and lets assume the time series is generated by the following stochastic process:
?Yt = ?Yt ?1 + u t . If the value of the estimator of ? is 0.7, and the variance of this estimator is 0.09, decide
based on the Dickey-Fuller test whether the stochastic process is stationary or non-stationary. The critical
value for applying the test for a significance level of 5% is -1,32; (1p)
b) Let
u? t be the residual variable values obtained from a simple regression model:
u? t : 1 -0.8
0.5
1
0.9
-0.8
-1
-1
0.6
-0.4
Verify the hypothesis of homoscedasticity for the residual variable based on a Fisher test. The values of
ordered based on the ascending values of
u? t are
X t , and critical F is F0.05; 3; 3 = 9.28 . (1p)
III (4p). We know that the returns on equities depend on the returm on the market. In this context, we present in the
following Table the return on equity, Y, for a fictional company and the return to the market, X, for 5 consecutive
months:
Table 1.
Luna
1
2
3
4
5
Return on equity for the fictional
company (%)
Return on the market (%)
5
3
6
6
5
8
5
9
7
6
For your convenience, here are some summary statistics: ?? ?? ?? = (
(
25
5
2
? 2
), ???
?? = 0.8, ???=1(???? ? ?? ) = 6.
181
5
35
35
5.1
), (?? ?? ??)?1 = (
255
?0.7
?0.7
), ?? ?? ?? =
0.1
Requirements:
a) Estimate and interpret the parameters
b)
? 0 and ? 1 (estimate using the OLS method); (0,5p)
Determine the confidence intervals for ? 0 and ? 1 for a confidence level of 95% ( t 0.025; 3 = 3.182 ); (1p)
c)
Test the hypothesis that the true parameter
? 1 is equal to 1.8 ( t 0.025; 3 = 3.182 ); (0,25p)
(
)
d) Test the validity of the model using the Fisher test F0.05;1;3 = 10.13 ; (0,75p)
e)
Calculate and interpret the coefficient of determination
(R ) ; (0,5p)
2
f) Predict the next month return on equity, if the market return will be 8%. Use interval estimators and predict both
the expected return on equity and the individual return on equity (the significance level considered is 5%). (1p)
Number II
I (3p). Let the following be a simultaneous equation model that present the demand and supply for a product based on
its price, the income and wealth of the consumers:
(D )
– demand function: Qt = ? 0 + ? 1 Pt + ? 2 I t + ? 3Wt + u1t
(S )
– supply function: Qt
= ? 0 + ?1 Pt + u 2t
(D )
= Qt( S )
– equilibrium relation: Qt
Requirements:
a) Present the types of variables and types of equalities in the model; (0,5p)
b) Write the model in the reduced form; (1p)
c) Identify each regression equation based on the order condition; (1p)
d) For each exactly- or over-identified equation describe the optimal method (methods) for estimating its
parameters. (0,5p)
II (2p).
a) Let
Yt be a time series and lets assume the time series is generated by the following stochastic process:
Yt = ?Yt ?1 + u t . If the value of the estimator of ? is 0.7, and the variance of this estimator is 0.09, decide
based on the Dickey-Fuller test whether the stochastic process is stationary or non-stationary. The critical
value for applying the test for a significance level of 5% is -1,32; (1p)
b) Let
u? t be the residual variable values obtained from a simple regression model:
u? t : 1 -0.8
0.5
1
0.9
-0.8
-1
-1
0.6
-0.4
Verify the hypothesis of non-autocorrelation for the residual variable based on a Durbin-Watson test (
d1 = 0.93 and d 2 = 1.33 for a significance level ? = 5% ). (1p)
III (4p). We know that the returns on equities depend on the returm on the market. In this context, we present in the
following Table the return on equity, Y, for a fictional company and the return to the market, X, for 5 consecutive
months:
Table 1.
Luna
Return on equity for the fictional
company (%)
1
2
3
4
5
Return on the market (%)
5
3
6
6
5
8
5
9
7
6
For your convenience, here are some summary statistics:
,
,
,
,
.
Requirements:
a) Estimate and interpret the parameters
b)
? 0 and ? 1 (estimate using the OLS method); (0,5p)
Determine the confidence intervals for ? 0 and ? 1 for a confidence level of 95% ( t 0.025; 3 = 3.182 ); (1p)
c)
Test the hypothesis that the true parameter
? 1 is equal to 2 ( t 0.025; 3 = 3.182 ); (0,25p)
(
)
d) Test the validity of the model using the Fisher test F0.05;1;3 = 10.13 ; (0,75p)
e)
f)
( )
2
Calculate and interpret the coefficient of determination R ; (0,5p)
Predict the next month return on equity, if the market return will be 10%. Use interval estimators and predict both
the expected return on equity and the individual return on equity (the significance level considered is 5%). (1p)
Number II
I (3p). Let the following be a simultaneous equation model that present the demand and supply for a product based on
its price, the income and wealth of the consumers:
(D )
– demand function: Qt = ? 0 + ? 1 Pt + ? 2 I t + ? 3Wt + u1t
(S )
– supply function: Qt
= ? 0 + ?1 Pt + u 2t
(D )
= Qt( S )
– equilibrium relation: Qt
Requirements:
a) Present the types of variables and types of equalities in the model; (0,5p)
b) Write the model in the reduced form; (1p)
c) Identify each regression equation based on the order condition; (1p)
d) For each exactly- or over-identified equation describe the optimal method (methods) for estimating its
parameters. (0,5p)
II (2p).
a) Let
Yt be a time series and lets assume the time series is generated by the following stochastic process:
Yt = ?Yt ?1 + u t . If the value of the estimator of ? is 0.7, and the variance of this estimator is 0.09, decide
based on the Dickey-Fuller test whether the stochastic process is stationary or non-stationary. The critical
value for applying the test for a significance level of 5% is -1,32; (1p)
b) Let
u? t be the residual variable values obtained from a simple regression model:
u? t : 1 -0.8
0.5
1
0.9
-0.8
-1
-1
0.6
-0.4
Verify the hypothesis of non-autocorrelation for the residual variable based on a Durbin-Watson test (
d1 = 0.93 and d 2 = 1.33 for a significance level ? = 5% ). (1p)
III (4p). We know that the returns on equities depend on the returm on the market. In this context, we present in the
following Table the return on equity, Y, for a fictional company and the return to the market, X, for 5 consecutive
months:
Table 1.
Luna
Return on equity for the fictional
company (%)
1
2
3
4
5
Return on the market (%)
5
3
6
6
5
8
5
9
7
6
For your convenience, here are some summary statistics:
,
,
,
,
.
Requirements:
a) Estimate and interpret the parameters
b)
? 0 and ? 1 (estimate using the OLS method); (0,5p)
Determine the confidence intervals for ? 0 and ? 1 for a confidence level of 95% ( t 0.025; 3 = 3.182 ); (1p)
c)
Test the hypothesis that the true parameter
? 1 is equal to 2 ( t 0.025; 3 = 3.182 ); (0,25p)
(
)
d) Test the validity of the model using the Fisher test F0.05;1;3 = 10.13 ; (0,75p)
e)
f)
( )
2
Calculate and interpret the coefficient of determination R ; (0,5p)
Predict the next month return on equity, if the market return will be 10%. Use interval estimators and predict both
the expected return on equity and the individual return on equity (the significance level considered is 5%). (1p)
Number I
I (3p). Let the following be a simultaneous equation model:
Ct = ? 0 + ? 1 I t + u1t
S t = ? 0 + ?1 I t + ? 2 I t ?1 + u 2t
I t = Ct + S t
where C t represents private consumption, I t – income, I t ?1 – income in the previous period (it is assumed
predetermined), S t represents savings.
Requirements:
a) Present the types of variables and types of equalities in the model; (0,5p)
b) Write the model in the reduced form; (1p)
c) Identify each regression equation based on the order condition; (1p)
d) For each exactly- or over-identified equation describe the optimal method (methods) for estimating its
parameters. (0,5p)
II (2p).
a) Let
Yt be a time series and lets assume the time series is generated by the following stochastic process:
?Yt = ?Yt ?1 + u t . If the value of the estimator of ? is 0.7, and the variance of this estimator is 0.09, decide
based on the Dickey-Fuller test whether the stochastic process is stationary or non-stationary. The critical
value for applying the test for a significance level of 5% is -1,32; (1p)
b) Let
u? t be the residual variable values obtained from a simple regression model:
u? t : 1 -0.8
0.5
1
0.9
-0.8
-1
-1
0.6
-0.4
Verify the hypothesis of homoscedasticity for the residual variable based on a Fisher test. The values of
ordered based on the ascending values of
u? t are
X t , and critical F is F0.05; 3; 3 = 9.28 . (1p)
III (4p). We know that the returns on equities depend on the returm on the market. In this context, we present in the
following Table the return on equity, Y, for a fictional company and the return to the market, X, for 5 consecutive
months:
Table 1.
Luna
1
2
3
4
5
Return on equity for the fictional
company (%)
Return on the market (%)
5
3
6
6
5
8
5
9
7
6
For your convenience, here are some summary statistics: ?? ?? ?? = (
(
25
5
2
? 2
), ???
?? = 0.8, ???=1(???? ? ?? ) = 6.
181
5
35
35
5.1
), (?? ?? ??)?1 = (
255
?0.7
?0.7
), ?? ?? ?? =
0.1
Requirements:
a) Estimate and interpret the parameters
b)
? 0 and ? 1 (estimate using the OLS method); (0,5p)
Determine the confidence intervals for ? 0 and ? 1 for a confidence level of 95% ( t 0.025; 3 = 3.182 ); (1p)
c)
Test the hypothesis that the true parameter
? 1 is equal to 1.8 ( t 0.025; 3 = 3.182 ); (0,25p)
(
)
d) Test the validity of the model using the Fisher test F0.05;1;3 = 10.13 ; (0,75p)
e)
Calculate and interpret the coefficient of determination
(R ) ; (0,5p)
2
f) Predict the next month return on equity, if the market return will be 8%. Use interval estimators and predict both
the expected return on equity and the individual return on equity (the significance level considered is 5%). (1p)
Number I
I (3p). Let the following be a simultaneous equation model:
Ct = ? 0 + ? 1 I t + u1t
S t = ? 0 + ?1 I t + ? 2 I t ?1 + u 2t
I t = Ct + S t
where C t represents private consumption, I t – income, I t ?1 – income in the previous period (it is assumed
predetermined), S t represents savings.
Requirements:
a) Present the types of variables and types of equalities in the model; (0,5p)
b) Write the model in the reduced form; (1p)
c) Identify each regression equation based on the order condition; (1p)
d) For each exactly- or over-identified equation describe the optimal method (methods) for estimating its
parameters. (0,5p)
II (2p).
a) Let
Yt be a time series and lets assume the time series is generated by the following stochastic process:
?Yt = ?Yt ?1 + u t . If the value of the estimator of ? is 0.7, and the variance of this estimator is 0.09, decide
based on the Dickey-Fuller test whether the stochastic process is stationary or non-stationary. The critical
value for applying the test for a significance level of 5% is -1,32; (1p)
b) Let
u? t be the residual variable values obtained from a simple regression model:
u? t : 1 -0.8
0.5
1
0.9
-0.8
-1
-1
0.6
-0.4
Verify the hypothesis of homoscedasticity for the residual variable based on a Fisher test. The values of
ordered based on the ascending values of
u? t are
X t , and critical F is F0.05; 3; 3 = 9.28 . (1p)
III (4p). We know that the returns on equities depend on the returm on the market. In this context, we present in the
following Table the return on equity, Y, for a fictional company and the return to the market, X, for 5 consecutive
months:
Table 1.
Luna
1
2
3
4
5
Return on equity for the fictional
company (%)
Return on the market (%)
5
3
6
6
5
8
5
9
7
6
For your convenience, here are some summary statistics: ?? ?? ?? = (
(
25
5
2
? 2
), ???
?? = 0.8, ???=1(???? ? ?? ) = 6.
181
5
35
35
5.1
), (?? ?? ??)?1 = (
255
?0.7
?0.7
), ?? ?? ?? =
0.1
Requirements:
a) Estimate and interpret the parameters
b)
? 0 and ? 1 (estimate using the OLS method); (0,5p)
Determine the confidence intervals for ? 0 and ? 1 for a confidence level of 95% ( t 0.025; 3 = 3.182 ); (1p)
c)
Test the hypothesis that the true parameter
? 1 is equal to 1.8 ( t 0.025; 3 = 3.182 ); (0,25p)
(
)
d) Test the validity of the model using the Fisher test F0.05;1;3 = 10.13 ; (0,75p)
e)
Calculate and interpret the coefficient of determination
(R ) ; (0,5p)
2
f) Predict the next month return on equity, if the market return will be 8%. Use interval estimators and predict both
the expected return on equity and the individual return on equity (the significance level considered is 5%). (1p)
1 of 1
1-1
1. Let the following be a simultaneous equation model:
C, = 2, +a1, +421
S, = Be + B.1, + B21,- +421
1, = C, +S,
where C, represents private consumption, I,
– income, 1-1
income in the previous period (it is assumed
predetermined), S, represents savings.
Requirements:
a) Present thetypes of variables and types ofequalities in the model;
b) Write the model in the reduced form”;
c) Identify each regression equation based on the order condition;
II.
Let Y, be a time series and let’s assume the time series is generated by the following stochastic process:
AY, = 89,41 +u, . If the value of the estimator of 8 is 0.7, and the variance of this estimator is 0.09,
decide based on the Dickey-Fuller test whether the stochastic process is stationary or non-stationary. The
critical value for applying the test fora significance level of 5% is -1,32.
1-1
III.
Let û, be the residual variable values obtained froma simple regression model:
û,: 1 -0.7
0.5 1 0.9 -0.9 -1 -1 0.6 -0.4
a) Verify the hypothesis ofhomoscedasticity forthe residual variable based on a Fishertest. The values of û,
are ordered based on the ascending values of X, and critical F” is F.05; 3;3 = 9.28.
b) Verify the hypothesis of non-autocorrelation for the residual variable based on a Durbin-Watson test (
d, = 0.93 and d = 1.33 for a significance level a = 5%).
::
IV. We know that the returns on equities depend on the returm on the market. In this context, we present in the following
Table the return on equity, Y, for a fictional company and the return to the market, X, for 5 consecutive months:
Luna
Return on the market (%)
Retum on equity forthe fictional
company (%)
6
9
5
7
8
4
5
5
=
XTY=
For your convenience, here are some summary statistics: XTX =
25
?
(35_255), (x+x)=+= (5:17 J.?)
(169), 0,2 = 0.8, 2X=1[Y,  7)2 = 6.
;3
Requirements:
a) Estimate and interpret the parameters B, and B, (estimate using the OLS method);
b) Determine the confidence intervals for B, and B, fora confidence levelof95%(10.025 = 3.182);
c) Test the hypothesis that the true parameter ß, is equalto 1.8(10.025; 3 = 3.182);
d) Test the validity of the modelusing the Fishertest (F.05.13 = 10.13);
e) Calculate and interpret the coefficient of determination (R²);
f) Predict the next month return on equity, if the market return will be 8%. Use intervalestimators and predict both the
expected retum onequity and the individual retum onequity (the significance level considered is 5%).