代写Time Series Modeling for Business代写C/C++编程

Time Series Modeling for Business

Part.1 True or  False:  Please  provide  an  interpretation  or  explanation  for your  answer; otherwise, no points will be awarded.

1.  In the finance literature, a stationary AR(1) model can be written as: rt  = (1 − φ1)μ + φ1rt−1  + at

Thus, the first order derivation of stationary AR(1) model is white noise, even if |φ1 | < 1.

2.  The response surface of multiple linear regression doesn’t have to be a straight line. In higher dimensional space, it can be a sphere.

3.  When we are doing regression, “unbiased” is the most important rule for modeling, otherwise the model will deviate from the center of data and residual will not be white noise.

4.  When using Monte Carlo simulation to estimate the variance of entire population. The equation should be n-1/1∑(Xi - X)
because Monte Carlo simulation is generating a sample to approximate the total population.

5.  Suppose you're on a game show, and you're given the choice of three doors: Behind one  door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who  knows what's behind the doors, opens another door, say No. 3, which has a goat. He then  says to you, "Do you want to pick door No. 2?" It is to your advantage to switch your choice.

6.  When using the AR(0) model to fit the data, the variance of the model is the variance of the residual.

7.  When applying the additive classical decomposition approach, after averaging the de- trended  series  by  season,  if  the  sum  of  the  seasonal  components  equals  1,  the seasonality has been successfully captured.

8.  In the Wiener process with drift model, the series mean and variance increase linearly with time t.

9.  A periodic time series is not stationary time series.

10. For two models, A and B, if model A has a lower AIC, it indicates that A is expected to provide better predictive performance, excluding long-term forecasting scenario.

Part 2: The following data represents the quarterly sales of small companies from 1981 to 2005. Please answer the questions.

a). Based on the plot provided, should you apply a multiplicative or additive model for classical decomposition? Should you opt for an AR, MA, or ARMA model? Provide the reasoning behind your choice. Can the order of the model be determined from above figures? If not, please explain why.

b). Following shows the R result for the model fitting.

Please write out the mathematical equation of the model based on the R result.

c). Interpret the following residual check results. What is the H0  to the p-value? Give conclusion on the model fitting.

d). Can you draw conclusion about the goodness of fit of the model from the residual checking result? Do you have any suggestions to improve the model?

Part3: Calculation

1.  Let the time series {rt }  follows an AR(2) series:

rt  = 0.75 − 0.5rt−1  + 0.5rt−2  + at , at  is white noise with σ  = 0.2

Determine if rt  is a stationary time series.

2.  One night, a taxi was involved in a hit-and-run accident. There are two taxi companies in  the city. One company's taxis are all green (the "Green" company), and it owns 85% of  the city's total taxis. The other company's taxis are all blue (the "Blue" company), and it  owns 15% of the city's total taxis. A witness claimed that the taxi involved in the accident  belonged to the "Blue" company. The court tested the witness's testimony and found that, under the circumstances at the time of the incident, the probability of the witness  correctly identifying the two colors was 80%. What is the probability that the taxi involved  in the accident was from the "Blue" company? (Express the answer as a percentage, ranging from 0% to 100%).

3.  Calculate the lag-2 covariance of time series

t	1	2	3	4	5	6	7
r	251	290	290	292	279	254	295