Predicting Stock Returns and Rate of Return Forecasting Models Project

Instructions:•
You must upload the work files from R (.prg). One for Part 1 and one for Part 2. Excel is
not suitable for this project, and it will not be accepted.
This project consists of two parts:
•
•
Part 1: Predicting Stock Returns.
Part 2: Forecasting models for the rate of inflation.
Part 1: Predicting Stock Returns.
Data Description:
Documentation for Stock_Returns_1931_2002
This file contains 2 monthly data series over the 1931:1-2002:12 sample period.
•
•
ExReturn: Excess Returns
ln_DivYield: 100×ln(dividend yield). (Multiplication by 100 means the changes are
interpreted as percentage points).
The data were supplied by Professor Motohiro Yogo of the University of Pennsylvania and were used in
his paper with John Campbell:
•
“Efficient Tests of Stock Return Predictability,” Journal of Financial Economics, 2006.
(Double click in the window below to access the data)
DATE_M
1931M01
1931M02
1931M03
1931M04
1931M05
1931M06
1931M07
1931M08
1931M09
EXRETURN
LN_DIVYIELD
5.9649584
-282.2329
10.3053054
-293.2089
-6.8408314
-287.8614
-10.4480653
-278.2477
-14.3580770
-265.4742
12.8502610
-280.5102
-6.6559179
-275.5950
0.0461485
-278.4424
-34.2583894
-247.1829
Some Background
exreturn: is the excess return on a broad-based index of stock prices, called the CRSP value-weighted
index, using monthly data from 1960:M1 to 2002:M12, where “M1” denotes the first month of the year
(January), “M2” denotes the second month, and so forth.
•
The monthly excess return is what you earn, in percentage terms, by purchasing a stock at the end
of the previous month and selling it at the end of this month, minus what you would have earned
had you purchased a safe asset (a U.S. Treasury bill). The return on the stock includes the capital
gain (or loss) from the change in price plus any dividends you receive during the month.
Calculating k-period stock returns:
One-period holding return:
𝑟𝑡,𝑡+1 = ln (
𝑟𝑡+1,𝑡+2 = ln (
𝑃𝑡+1
) = ln(𝑃𝑡+1 ) − ln(𝑃𝑡 ) = 𝑝𝑡+1 − 𝑝𝑡 = Δ𝑝𝑡+1
𝑃𝑡
𝑃𝑡+2
) = ln(𝑃𝑡+2 ) − ln(𝑃𝑡+1 ) = 𝑝𝑡+2 − 𝑝𝑡+1 = Δ𝑝𝑡+2
𝑃𝑡+1
Two-period holding return:
𝑃𝑡+2
𝑟𝑡,𝑡+2 = ln (
) = ln(𝑃𝑡+2 ) − ln(𝑃𝑡 ) = 𝑝𝑡+2 − 𝑝𝑡 = 𝑝𝑡+2 − 𝑝𝑡+1 + 𝑝𝑡+1 − 𝑝𝑡
𝑃𝑡
= Δ𝑝𝑡+2 + Δ𝑝𝑡+1 = 𝑟𝑡+1,𝑡+2 + 𝑟𝑡,𝑡+1
Other way
𝑟𝑡,𝑡+2 = ln (
𝑃𝑡+2
𝑃𝑡+2 𝑃𝑡+1
𝑃𝑡+1 𝑃𝑡+2
𝑃𝑡+1
𝑃𝑡+2
) = ln (
) = ln (
) = ln (
) + ln (
)
𝑃𝑡
𝑃𝑡 𝑃𝑡+1
𝑃𝑡 𝑃𝑡+1
𝑃𝑡
𝑃𝑡+1
= 𝑟𝑡,𝑡+1 + 𝑟𝑡+1,𝑡+2
Three-period’s returns:
𝑟𝑡,𝑡+3 = ln (
𝑟𝑡,𝑡+3 = ln (
𝑃𝑡+3
) = ln(𝑃𝑡+3 ) − ln(𝑃𝑡 ) = 𝑝𝑡+3 − 𝑝𝑡
𝑃𝑡
𝑃𝑡+3
𝑃𝑡+3 𝑃𝑡+2 𝑃𝑡+1
𝑃𝑡+1 𝑃𝑡+2 𝑃𝑡+3
) = ln (
) = ln (
)
𝑃𝑡
𝑃𝑡 𝑃𝑡+2 𝑃𝑡+1
𝑃𝑡 𝑃𝑡+1 𝑃𝑡+2
𝑃𝑡+1
𝑃𝑡+2
𝑃𝑡+3
= ln (
) + ln (
) + ln (
) = 𝑟𝑡,𝑡+1 + 𝑟𝑡+1,𝑡+2 + 𝑟𝑡+2,𝑡+3
𝑃𝑡
𝑃𝑡+1
𝑃𝑡+2
k-period’s returns:
𝑟𝑡,𝑡+𝑘 = ln (
𝑟𝑡,𝑡+𝑘 = ln (
𝑃𝑡+𝑘
) = ln(𝑃𝑡+𝑘 ) − ln(𝑃𝑡 ) = 𝑝𝑡+𝑘 − 𝑝𝑡
𝑃𝑡
𝑃𝑡+𝑘
𝑃𝑡+𝑘 𝑃𝑡+𝑘−1
𝑃𝑡+2 𝑃𝑡+1
𝑃𝑡+1 𝑃𝑡+2 𝑃𝑡+3
𝑃𝑡+𝑘
) = ln (
⋅ …⋅
) = ln (
⋅ …⋅
)
𝑃𝑡
𝑃𝑡 𝑃𝑡+𝑘−1
𝑃𝑡+2 𝑃𝑡+1
𝑃𝑡 𝑃𝑡+1 𝑃𝑡+2
𝑃𝑡+𝑘−1
𝑃𝑡+1
𝑃𝑡+2
𝑃𝑡+3
𝑃𝑡+𝑘
= ln (
) + ln (
) + ln (
) + ⋯ + ln (
)
𝑃𝑡
𝑃𝑡+1
𝑃𝑡+2
𝑃𝑡+𝑘−1
= 𝑟𝑡,𝑡+1 + 𝑟𝑡+1,𝑡+2 + 𝑟𝑡+2,𝑡+3 + ⋯ + 𝑟𝑡+𝑘−1,𝑡+𝑘
When to apply a “buy and hold” strategy:
•
•
•
•
•
If you have a reliable “forecast” of future stock returns then an active “buy and hold” strategy
will make you rich quickly by beating the stock market.
If you think that the stock market will be going up, you should buy stocks today and sell them
later, before the market turns down. Forecasts based on past values of stock returns are sometimes
called “momentum” forecasts: If the value of a stock rose this month, perhaps it has momentum
and will also rise next month.
If so, then returns will be autocorrelated, and the autoregressive model will provide useful
forecasts. You can implement a momentum-based strategy for a specific stock or for a stock
index that measures the overall value of the market.
From another point of view, we can use autoregressive models to test a version of the efficient
markets hypothesis (EMH). A strict form of the efficient markets hypothesis states that information
observable to the market prior to period 𝑡 should not help to predict the return during period 𝑡. If
the (EMH) is false, then returns might be predictable. If so, then returns will be autocorrelated, and
the autoregressive model will provide useful forecasts.
For example, if you want to find out if returns are predictable (even if it is just a bit), estimate the
following AR(1)
𝑅𝑡 = 𝛽0 + 𝛽1 𝑅𝑡−1 + 𝑢𝑡+1
•
•
•
A positive 𝛽1 coefficient means “momentum,” past “good returns” mean higher future returns.
A negative 𝛽1 coefficient means “overreaction” or “mean reversion”. In this case, previous
“good returns” mean lower future returns.
Either way, if 𝛽1 ≠ 0, then returns will be autocorrelated, and the autoregressive model will
provide useful forecasts.
Note: In all your calculations use Huber-White heteroskedasticity consistent standard errors and
covariance.
a. Repeat the calculations reported in Table 15.2, using the following regression specifications
estimated over the 1960:M1–2002:M12 sample period.
AR(1) Model
𝑟𝑡 = 𝛽0 + 𝛽1 𝑟𝑡−1 + 𝑒𝑡
AR(2) Model
𝑟𝑡 = 𝛽0 + 𝛽1 𝑟𝑡−1 + 𝛽2 𝑟𝑡−2 + 𝑒𝑡
AR(4) Model
𝑟𝑡 = 𝛽0 + 𝛽1 𝑟𝑡−1 + 𝛽2 𝑟𝑡−2 + 𝛽3 𝑟𝑡−3 + 𝛽4 𝑟𝑡−4 + 𝑒𝑡
Autoregressive Models of Monthly Excess Stock Returns, 1960:M1–2002:M12
Dependent variable: Excess returns on the CRSP value-weighted index
Specification
Regressors
(1)
AR(1)
(2)
AR(2)
(3)
AR(4)
Excess Ret(t-1)
Std. Error
p-value
Excess Ret(t-2)
Std. Error
p-value
Excess Ret(t-3)
Std. Error
p-value
Excess Ret(t-4)
Std. Error
p-value
Intercept
Std. Error
p-value
Adj R^2
Wald F-statistic
p-value
T=
b. Are these results consistent with the theory of efficient capital markets?
c. Can you provide an intuition behind this result?
d. Repeat the calculations reported in Table 15.6, using regressions estimated over the 1960:M1–
2002:M12 sample period.
Autoregressive Distributed Lag Models of Monthly Excess Stock Returns, 1960:M1–2002:M12
Dependent variable: Excess returns on the CRSP value-weighted index
(1)
(2)
Specification
ADL(1,1)
ADL(2,2)
Eatimation Period
1960:M1–2002:M12
1960:M1–2002:M12
Regressors
Excess Ret(t-1)
Std. Error
p-value
(3)
ADL(1,1)
1960:M1–1992:M12
Excess Ret(t-2)
Std. Error
p-value
Change_ln_DP(t-1)
Std. Error
p-value
Change_ln_DP(t-2)
Std. Error
p-value
ln_DP(t-1)
Std. Error
p-value
Intercept
Std. Error
p-value
Adj R^2
F-statistic
p-value
Obs =
e. Does the Δln(dividend yield) have any predictive power for stock returns?
f.
Does “the level of the dividend yield” have any predictive power for stock returns?
g. Construct pseudo out-of-sample forecasts of excess returns over the 1993:M1–2002:M12 period,
using the regression specifications below that begin in 1960:M1.
ADL(1,1) specification:
𝐸𝑅𝑡 = 𝛽0 + 𝛽1 𝐸𝑅𝑡−1 + 𝛽2 𝑙𝑛(𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑)𝑡−1 + 𝑢𝑡
Constant Forecast: (in which the recursively estimated forecasting model includes only an
intercept)
𝐸𝑅𝑡 = 𝛽0 + 𝑢𝑡
Zero Forecast: the sample RMSFEs of always forecasting excess returns to be zero.
Model
Zero Forecast
RMSFE
Constant Forecast
ADL(1, 1)
h. Does the ADL(1,1) model with the log dividend yield provide better forecasts than the zero or
constant models?
Part 2
Forecasting models for the rate of inflation – Guidelines
Go to FRED’s website (https://fred.stlouisfed.org/) and download the data for:
•
Consumer Price Index for All Urban Consumers: All Items (CPIAUCSL) – Seasonally adjusted –
Monthly Frequency – From 1947:M1 to 2017:M12
In this hands-on exercise you will construct forecasting models for the rate of inflation, based on
CPIAUCSL.
For this analysis, use the sample period 1970:M01–2012:M12 (where data before 1970 should be used, as
necessary, as initial values for lags in regressions).
a.
(i)
Compute the (annualized) inflation rate,
(ii)
Plot the value of Infl from 1970:M01 through 2012:M12. Based on the plot, do you think
that Infl has a stochastic trend? Explain.
(i)
Compute the first twelve autocorrelations of (𝐼𝑛𝑓𝑙 𝑎𝑛𝑑 Δ𝐼𝑛𝑓𝑙)
(ii)
Plot the value of Δ𝐼𝑛𝑓𝑙 from 1970:M01 through 2012:M12. The plot should look
“choppy” or “jagged.” Explain why this behavior is consistent with the first
autocorrelation that you computed in part (i) for Δ𝐼𝑛𝑓𝑙.
(i)
Compute Run an OLS regression of 𝐼𝑛𝑓𝑙𝑡 on 𝐼𝑛𝑓𝑙𝑡−1. Does knowing the inflation this
month help predict the inflation next month? Explain.
(ii)
Estimate an AR(2) model for Infl. Is the AR(2) model better than an AR(1) model?
Explain.
(iii)
Estimate an AR(p) model for 𝑝 = 0, … ,8. What lag length is chosen by BIC? What lag
length is chosen by AIC?
(iv)
Use the AR(2) model to predict “the level of the inflation rate” in 2013:M01—that is, 𝐼𝑛𝑓𝑙2013:𝑀01 .
(i)
Use the ADF test for the regression in Equation (14.31) with two lags of 𝛥𝐼𝑛𝑓𝑙 to test for
a stochastic trend in 𝐼𝑛𝑓𝑙.
b.
c.
d.
(ii)
Is the ADF test based on Equation (14.31) preferred to the test based on Equation (14.32)
for testing for stochastic trend in 𝐼𝑛𝑓𝑙? Explain.
(iii)
In (i) you used two lags of 𝛥𝐼𝑛𝑓𝑙. Should you use more lags? Fewer lags? Explain.
(iv)
Based on the test you carried out in (i), does the AR model for 𝐼𝑛𝑓 contain a unit root?
Explain carefully. (Hint: Does the failure to reject a null hypothesis mean that the null
hypothesis is true?)
e. Use the QLR test with 15% trimming to test the stability of the coefficients in the AR(2) model
for “the inflation” 𝐼𝑛𝑓𝑙. Is the AR(2) model stable? Explain.
f.
(i)
Using the AR(2) model for 𝐼𝑛𝑓𝑙 with a sample period that begins in 1970:M01, compute
pseudo out-of-sample forecasts for the inflation beginning in 2005:M12 and going
through 2012:M12.
(ii)
Are the pseudo out-of-sample forecasts biased? That is, do the forecast errors have a
nonzero mean?
(iii)
How large is the RMSFE of the pseudo out-of-sample forecasts? Is this consistent with
the AR(2) model for 𝐼𝑛𝑓𝑙 estimated over the 1970:M01–2005:M12 sample period?
(iv)
There is a large outlier in 2008:Q4. Why did inflation fall so much in 2008:Q4? (Hint:
Collect some data on oil prices. What happened to oil prices during 2008?)
Instructions:
• Please complete the guided project by May 8, 11:59 PM (ET).
• Write your answer below each question and upload a “word doc” named
LastName_FirstName.doc using the link on Blackboard.
• Also, you must upload the work files from R (LastName_FirstName.prg). One for
Part 1 and one for Part 2. Excel is not suitable for this project and it will not be
accepted.
This project consist of two parts:
• Part 1: Predicting Stock Returns.
• Part 2: Forecasting models for the rate of inflation.
Part 1: Predicting Stock Returns.
Data Description:
Documentation for Stock_Returns_1931_2002
This file contains 2 monthly data series over the 1931:1-2002:12 sample period.
ExReturn: Excess Returns
ln_DivYield (growth rate): 100×ln(dividend yield). (Multiplication by 100 means the
changes are interpreted as percentage points).
The data were supplied by Professor Motohiro Yogo of the University of Pennsylvania and
were used in his paper with John Campbell:
“Efficient Tests of Stock Return Predictability,” Journal of Financial
Economics, 2006. (Double click in the window below to access the data)
DATE_M
1931M01
1931M02
1931M03
1931M04
1931M05
1931M06
1931M07
1931M08
1931M09
EXRETURN
LN_DIVYIELD
5.9649584
-282.2329
10.3053054
-293.2089
-6.8408314
-287.8614
-10.4480653
-278.2477
-14.3580770
-265.4742
12.8502610
-280.5102
-6.6559179
-275.5950
0.0461485
-278.4424
-34.2583894
-247.1829
Some Background
exreturn: is the excess return on a broad- based index of stock prices, called
the CRSP value -weighted index, using monthly data from 1960:M1 to
2002:M12, where “M1” denotes the first month of the year (January), “M2”
denotes the second month, and so forth.
•
The monthly excess return is what you earn, in percentage terms,
by purchasing a stock at the end of the previous month and selling
it at the end of this month, minus what you would have earned had
you purchased a safe asset (a U.S. Treasury bill). The return on the
stock includes the capital gain (or loss) from the change in price
plus any dividends you receive during the month.
Calculating k-period stock returns:
One-period holding return:
Two-period holding return:
Other way
Three-period’s returns:
k-period’s returns:
When to apply a “buy and hold” strategy:
•
•
•
•
•
If you have a reliable “forecast” of future stock returns then an active “buy and hold”
strategy will make you rich quickly by beating the stock market.
If you think that the stock market will be going up, you should buy stocks today and sell
them later, before the market turns down. Forecasts based on past values of stock returns
are sometimes called “momentum” forecasts: If the value of a stock rose this month,
perhaps it has momentum and will also rise next month.
If so, then returns will be autocorrelated, and the autoregressive model will provide
useful forecasts. You can implement a momentum-based strategy for a specific stock or
for a stock index that measures the overall value of the market.
From another point of view, we can use autoregressive models to test a version of the
efficient markets hypothesis (EMH). A strict form of the efficient markets hypothesis
states that information observable to the market prior to period should not help to
predict the return during period . If the (EMH) is false, then returns might be
predictable. If so, then returns will be autocorrelated, and the autoregressive model will
provide useful forecasts.
For example, if you want to find out if returns are predictable (even if it is just a bit),
estimate the following AR(1)
•
•
•
A positive
coefficient means “momentum,” past “good returns” mean higher
future returns.
A negative
coefficient means “overreaction” or “mean reversion”. In this case,
previous “good returns” mean lower future returns.
Either way, if
, then returns will be autocorrelated, and the autoregressive
model will provide useful forecasts.
Note: In all your calculations use Huber-White heteroskedasticity consistent standard errors and
covariance.
a. Repeat the calculations reported in Table 14.2, using the following regression
specifications estimated over the 1960:M1–2002:M12 sample period.
AR(1) Model
AR(2) Model
AR(4) Model
Autoregressive Models of Monthly Excess Stock Returns, 1960:M1–
2002:M12
Dependent variable: Excess returns on the CRSP value-weighted
index
(1)
(2)
(3)
Specification
AR(1)
AR(2)
AR(4)
Regressors
Excess Ret(t-1)
0.05
0.053
0.054
Std. Error
0.051
0.051
0.051
p-value
0.327
0.297
0.297
Excess Ret(t-2)
Std. Error
p-value
-0.053
0.048
0.274
-0.054
0.048
0.265
Excess Ret(t-3)
Std. Error
p-value
0.009
0.05
0.853
Excess Ret(t-4)
Std. Error
p-value
-0.016
0.047
0.736
Intercept
Std. Error
p-value
0.312
0.198
0.116
0.328
0.199
0.1004
0.33
0.203
0.105
Adj R^2
0.001
0.001
-0.002
Wald F-statistic
p-value
0.965
0.327
1.334
0.264
0.7
0.592
T=
516
516
516
b. Are these results consistent with the theory of efficient capital markets?
Yes, these results are consistent with the theory of efficient capital market because the P-value in
Excess Ret(t-1) is greater than 0.05. This shows that the AR (1) model is hardly useful in
forecasting. It is also greater than 0.05 in AR (2) also indicating that the respective models are
not useful for forecasting. However, it is less than 0.05 significant level for both Excess Ret(t-3)
at 0.009 and Excess Ret(t-4) at 0.016. These results are also consistent with the theory because it
holds that excess returns can be unpredictable.
c. Can you provide an intuition behind this result?
An intuition behind the results is that the autoregressive model of the monthly excess stock
returns is hardly a good predictor of stock price returns. The results are in line with efficient
financial markets hypothesis which holds that there are no statistically significant coefficients in
the estimated models. In this regard, we cannot reject the hypotheses that the coefficients are
equal to zero. Since the Adjustment R2 is almost to zero in the three models with a negative for
AR(4) model, we can resonate that the models are not good predictors of stock returns.
d. Repeat the calculations reported in Table 14.6, using regressions estimated over the
1960:M1– 2002:M12 sample period.
Autoregressive Distributed Lag Models of Monthly Excess Stock Returns, 1960:M1–
2002:M12
Dependent variable: Excess returns on the CRSP value-weighted index
(1)
(2)
Specification
ADL(1,1)
ADL(2,2)
Eatimation Period
1960:M1–2002:M12
1960:M1–2002:M12
Regressors
Excess Ret(t-1)
0.059
0.042
Std. Error
0.158
0.162
p-value
0.71
0.798
Excess Ret(t-2)
Std. Error
p-value
Change_ln_DP(t-1)
Std. Error
p-value
(3)
ADL(1,1)
1960:M1–1992:M12
0.078
0.057
0.17
-0.213
0.193
0.237
0.009
0.157
0.957
Change_ln_DP(t-2)
Std. Error
p-value
-0.012
0.163
0.944
-0.161
0.185
0.386
ln_DP(t-1)
Std. Error
p-value
0.026
0.012
0.025
Intercept
Std. Error
p-value
0.309
0.199
0.123
0.372
0.208
0.076
8.987
3.912
0.015
Adj R^2
-0.001
-0.001
0.013
F-statistic
p-value
Obs =
0.625
0.521
516
0.897
0.466
516
3.683
0.026
396
e. Does the
have any predictive power for stock returns?
The
lacks the predictive power for stock returns since the p-values are
higher than 0.05. This means that the null hypothesis is not rejected. Typically, the models of the
t-statistics tend to point towards the notion that the respective coefficients are not zero
f. Does “the level of the dividend yield” have any predictive power for stock returns?
Yes, “the level of the dividend yield” has predictive power for stock returns. The p-value
is 0, which necessitates us to reject the null hypothesis.
g. Construct pseudo out-of-sample forecasts of excess returns over the 1993:M1–
2002:M12 period, using the regression specifications below that begin in 1960:M1.
ADL(1,1) specification:
Constant Forecast: (in which the recursively estimated forecasting model includes only an
intercept)
Zero Forecast: the sample RMSFEs of always forecasting excess returns to be zero.
Model
RMSFE
Zero Forecast
3.99
Constant Forecast
3.98
ADL(1, 1)
4.30
h. Does the ADL(1,1) model with the log dividend yield provide better forecasts than
the zero or constant models?
I would say no. The ADL(1,1) model with the log dividend yield does not provide better
forecasts than the zero or constant models. With regards to the results, the model performs
poorer based on the RMSFE sample with respect to the intercept-only function. A model
forecasting that excess returns would be zero would further register a lower sample RMSFE.
Part 2
Forecasting models for the rate of inflation – Guidelines
Go to FRED’s website (https://fred.stlouisfed.org/) and download the data for:
Consumer Price Index for All Urban Consumers: All Items (CPIAUCSL) – Seasonally
adjusted – Monthly Frequency – From 1947:M1 to 2017:M12
In this hands-on exercise you will construct forecasting models for the rate of inflation,
based on CPIAUCSL.
For this analysis, use the sample period 1970:M01–2012:M12 (where data before 1970 should
be used, as necessary, as initial values for lags in regressions).
a.
(i)
Compute the (annualized) inflation rate.
(ii)
22.445
Plot the value of Infl from 1970:M01 through 2012:M12. Based on the plot, do
you think that Infl has a stochastic trend? Explain.
Based on the plot, it is probable that inflation has a stochastic trend since the
trend for the next day can hardly be predicted. This can only mean that the
trend is statistically random.
b.
(i)
Compute the first twelve autocorrelations of
From R-results
(ii)
Plot the value of
from 1970:M01 through 2012:M12. The plot should look
“choppy” or “jagged.” Explain why this behavior is consistent with the first
autocorrelation that you computed in part (i) for
.
The behavior is consistent with the autocorrelation that I computed in part (i) for
because
the longer bars towards the edges for some years depict higher volatility compared to the shorter
bars in the middle which depict stability.
c.
(i)
Compute Run an OLS regression of
on
. Does knowing the
inflation this month help predict the inflation next month? Explain.
Based on the results, it is imparative to agree that knowing the inflation in this
month helps predict the inflation in next month. This follows the fact that p =
0, and hence the null hypothesis is worth rejecting. This means that there is
predictive power for the next month.
(ii)
Estimate an AR(2) model for Infl. Is the AR(2) model better than an AR(1)
model? Explain.
Based on the results, the AR(2) model is much better than an AR(1) model because the Rsquare for AR(2) is greater than that of AR(1).
(iii) Estimate an AR(p) model for p =0, …, 8 . What lag length is chosen by BIC?
What lag length is chosen by AIC?
From R-results: Both AIC and BIC have lag lengths of 2 each.
d.
(i)
Use the AR(2) model to predict “the level of the inflation rate” in 2013:M01—
that is,
.
From R-results, the level of the inflation rate in 2013 is 0.017743 or 1.77%.
(ii)
Is the ADF test based on Equation (14.31)
preferred to the test based on Equation (14.32) for testing for stochastic trend in
? Explain.
From R-results, we can conclude that it is preferred since when the p-value is increases, it will
not be significant once the lag period is add and it is not significant.
(iii)
In (i) you used two lags of
. Should you use more lags? Fewer lags?
Explain.
More lags should not be used because it is not possible to add more variables owing to the fact
that the value is not that much significant.
(iv)
Based on the test you carried out in (i), does the AR model for
contain a unit
root? Explain carefully. (Hint: Does the failure to reject a null hypothesis mean
that the null hypothesis is true?)
Based on the test in (i), the AR model for
does not contain a unit root since the t-statistic in
this case is -2.3 and the p = 0.1724. This means that the null hypothesis cannot be rejected.
e. Use the QLR test with 15% trimming to test the stability of the coefficients in the
AR(2) model for “the inflation” Infl . Is the AR(2) model stable? Explain.
Based on the R-results, we can ascertain that the AR(2) model is not stable because the QLR is
29.7 with a p-critical value of about 6.02. This implies that we should reject the null hypothesis
since at least one of the coefficients changes over time.
f.
(i) Using the AR(2) model for
with a sample period that begins in 1970:M01, compute
pseudo out-of-sample forecasts for the inflation beginning in 2005:M12 and going
through 2012:M12.
(ii) Are the pseudo out-of-sample forecasts biased? That is, do the forecast errors have a
nonzero mean?
I would say yes that the pseudo out-of-sample forecasts are biased. The biasness value is 0.279.
(iii)
How large is the RMSFE of the pseudo out-of-sample forecasts? Is this consistent
with the AR(2) model for
estimated over the 1970:M01–2005:M12 sample
period?
Accordingly, RMSFE is 0.020259 and is consistent with the AR(2) model for
over the 1970:M01–2005:M12 sample period.
(iv)
estimated
There is a large outlier in 2008:Q4. Why did inflation fall so much in
2008:Q4? (Hint: Collect some data on oil prices. What happened to oil prices
during 2008?)
From R-results, the rate of inflation fell significantly 2008. Inflation fell so much because of
the 2008 financial crisis and its impact of the global economy, which lead to the witnessed
economic decline as evident through a serious drop in oil prices.
TABLE 15.2 Autoregressive Models of Monthly Excess Stock Returns, 1960:M1–2002:M12
Table 15.6 Autoregressive Distributed Lag Models of Monthly Excess Stock Returns