Reading 13- LOS l : Q1- 4 style, black, class

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1．he data below yields the following AR(1) specification: x_t = 0.9 – 0.55x_t-1 + E_t , and the indicated fitted values and residuals.

Time	xt	fitted values	residuals
1	1	-	-
2	-1	0.35	-1.35
3	2	1.45	0.55
4	-1	-0.2	-0.8
5	0	1.45	-1.45
6	2	0.9	1.1
7	0	-0.2	0.2
8	1	0.9	0.1
9	2	0.35	1.65

The following sets of data are ordered from earliest to latest. To test for ARCH, the researcher should regress:

A)   (0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01, 2.7225) on (1.8225, 0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01).

B)   (1, 4, 1, 0, 4, 0, 1, 4) on (1, 1, 4, 1, 0, 4, 0, 1)

C)   (0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01, 2.7225) on (1, 1, 4, 1, 0, 4, 0, 1)

D)   (-1.35, 0.55, -0.8, -1.45, 1.1, 0.2, 0.1, 1.65) on (0.35, 1.45, -0.2, 1.45, 0.9, -0.2, 0.9, 0.35)

2．ppose you estimate the following model of residuals from an autoregressive model:

ε_t² = 0.4 + 0.80ε_t-1² + µ_t, where ε = ε^{^}

If the residual at time t is 2.0, the forecasted variance for time t+1 is:

A)   3.2.

B)   3.6.

C)   2.0.

D)   1.2.

3．ppose you estimate the following model of residuals from an autoregressive model:

ε_t² = 0.25 + 0.6ε²_t-1 + µ_t, where ε = ε^{^}

If the residual at time t is 0.9, the forecasted variance for time t+1 is:

A)   0.736.

B)   0.790.

C)   0.850.

D)   0.819.

4．ich of the following is NOT a consequence of a model containing ARCH(1) errors? The:

A)   regression parameters will be incorrect.

B)   variance of the errors can be predicted.

C)   standard error of the regression coefficients will be incorrect.

D)   model's specification can be corrected by adding an additional lag variable.

spaceedu

1．he data below yields the following AR(1) specification: x_t = 0.9 – 0.55x_t-1 + E_t , and the indicated fitted values and residuals.

Time	xt	fitted values	residuals
1	1	-	-
2	-1	0.35	-1.35
3	2	1.45	0.55
4	-1	-0.2	-0.8
5	0	1.45	-1.45
6	2	0.9	1.1
7	0	-0.2	0.2
8	1	0.9	0.1
9	2	0.35	1.65

The following sets of data are ordered from earliest to latest. To test for ARCH, the researcher should regress:

A)   (0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01, 2.7225) on (1.8225, 0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01).

B)   (1, 4, 1, 0, 4, 0, 1, 4) on (1, 1, 4, 1, 0, 4, 0, 1)

C)   (0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01, 2.7225) on (1, 1, 4, 1, 0, 4, 0, 1)

D)   (-1.35, 0.55, -0.8, -1.45, 1.1, 0.2, 0.1, 1.65) on (0.35, 1.45, -0.2, 1.45, 0.9, -0.2, 0.9, 0.35)

The correct answer was A)

Heteroskedasticity describes one possible pattern of the squared residuals. The ARCH model is the regression of the squared residuals on their corresponding lagged values. The squared residuals are (1.8225, 0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01, 2.7225). Regressing the last 7 on the first 7 would be a first-order ARCH model. Regressing the squared residuals on xt, i.e., (0.3025, 0.64, 2.1025, 1.21, 0.04, 0.01, 2.7225) on (1, 1, 4, 1, 0, 4, 0, 1), would be a test for another type of conditional heteroskedasticity, but not ARCH.

2．ppose you estimate the following model of residuals from an autoregressive model:

ε_t² = 0.4 + 0.80ε_t-1² + µ_t, where ε = ε^{^}

If the residual at time t is 2.0, the forecasted variance for time t+1 is:

A)   3.2.

B)   3.6.

C)   2.0.

D)   1.2.

The correct answer was B)

The variance at t=t+1 is 0.4 + [0.80 (4.0)] = 0.4 + 3.2. = 3.6.

3．ppose you estimate the following model of residuals from an autoregressive model:

ε_t² = 0.25 + 0.6ε²_t-1 + µ_t, where ε = ε^{^}

If the residual at time t is 0.9, the forecasted variance for time t+1 is:

A)   0.736.

B)   0.790.

C)   0.850.

D)   0.819.

The correct answer was A)

The variance at t=t+1 is 0.25 + [0.60 (0.81)] = 0.25 + 0.486 = 0.736.

4．ich of the following is NOT a consequence of a model containing ARCH(1) errors? The:

A)   regression parameters will be incorrect.

B)   variance of the errors can be predicted.

C)   standard error of the regression coefficients will be incorrect.

D)   model's specification can be corrected by adding an additional lag variable.

The correct answer was D)

The presence of autoregressive conditional heteroskedasticity (ARCH) indicates that the variance of the error terms is not constant. This is a violation of the regression assumptions upon which time series models are based. The addition of another lag variable to a model is not a means for correcting for ARCH (1) errors.