Question

In: Statistics and Probability

This data shows the series of quarterly shipments in millions of US dollars of US hosehold appliances between 1985 and 1989.

Please show answers and code using R/R Studio.

This data shows the series of quarterly shipments in millions of US dollars of US hosehold appliances between 1985 and 1989.

Quarter Shipments
Q1-1985 4009
Q1-1986 4123
Q1-1987 4493
Q1-1988 4595
Q1-1989 4245
Q2-1985 4321
Q2-1986 4522
Q2-1987 4806
Q2-1988 4799
Q2-1989 4900
Q3-1985 4224
Q3-1986 4657
Q3-1987 4551
Q3-1988 4417
Q3-1989 4585
Q4-1985 3944
Q4-1986 4030
Q4-1987 4485
Q4-1988 4258
Q4-1989 4533

(b) apply a moving average with window span w=4 to the data. Use all but the last year as the training period. Create a time ploy of the moving average series.

- What does the MA (4) chart reveal?

- Use the MA (4) model to forecast appliance sales in Q1-1990

-  Use the MA (4) model to forecast appliance sales in Q1-1991

- is the Q1-1990 forecast most likely to underestimate, overestimate, or accurately estimate the actual sales on Q1-1990? Explain?

- Managment Feels most comfortable with moving averages. the analyst therefore plans to use this method for forecasting future quaters. what else should be considered before using the MA(4) to forecast future quarterly shipments of household appliances.

(c) we now focus on forcasting beyond 1989. in the following continue to use all but the last year as the training period and the last four quarters as the validation period. apply holt winters exponential smoothing to the training period.

- compute MAPE values for the training and validation periods using holt winters exponential smoothing.

- Draw two time plots: one for the actual forecasted values and theo ther for the residuals. the x-acis should include the training and validation periods. comment on the model fit in the training and validation periods.

- if we optimize the smoothing constants in the holt winters method are the optimal values likely to be close to zero? why or why not?

Solutions

Expert Solution

(b)

Qtr Shipments Period Shipments Forecast (MA4)
Q1-1985 4009 1 4009
Q1-1986 4123 2 4321
Q1-1987 4493 3 4224
Q1-1988 4595 4 3944
Q1-1989 4245 5 4123 4124.5
Q2-1985 4321 6 4522 4153.0
Q2-1986 4522 7 4657 4203.3
Q2-1987 4806 8 4030 4311.5
Q2-1988 4799 9 4493 4333.0
Q2-1989 4900 10 4806 4425.5
Q3-1985 4224 11 4551 4496.5
Q3-1986 4657 12 4485 4470.0
Q3-1987 4551 13 4595 4583.8
Q3-1988 4417 14 4799 4609.3
Q3-1989 4585 15 4417 4607.5
Q4-1985 3944 16 4258 4574.0
Q4-1986 4030 17 4245 4517.3
Q4-1987 4485 18 4900 4429.8
Q4-1988 4258 19 4585 4455.0
Q4-1989 4533 20 4533 4497.0
Q1-1990 21 4565.8
Q1-1991 22 4565.8

For estimating the over/ underestimate, we will find the value of the tracking signal by dividing the sum of errors by the MAD.

Qtr Shipments Period Shipments Forecast (MA4) |At - Ft| (At - Ft)
Q1-1985 4009 1 4009
Q1-1986 4123 2 4321
Q1-1987 4493 3 4224
Q1-1988 4595 4 3944
Q1-1989 4245 5 4123 4124.5 1.5 -1.5
Q2-1985 4321 6 4522 4153.0 369 369.0
Q2-1986 4522 7 4657 4203.3 453.75 453.8
Q2-1987 4806 8 4030 4311.5 281.5 -281.5
Q2-1988 4799 9 4493 4333.0 160 160.0
Q2-1989 4900 10 4806 4425.5 380.5 380.5
Q3-1985 4224 11 4551 4496.5 54.5 54.5
Q3-1986 4657 12 4485 4470.0 15 15.0
Q3-1987 4551 13 4595 4583.8 11.25 11.3
Q3-1988 4417 14 4799 4609.3 189.75 189.8
Q3-1989 4585 15 4417 4607.5 190.5 -190.5
Q4-1985 3944 16 4258 4574.0 316 -316.0
Q4-1986 4030 17 4245 4517.3 272.25 -272.3
Q4-1987 4485 18 4900 4429.8 470.25 470.3
Q4-1988 4258 19 4585 4455.0 130 130.0
Q4-1989 4533 20 4533 4497.0 36 36.0
Q1-1990 21 4565.8 208.2344 5.802356
Q1-1991 22 4565.8 MAD TS

Since the tracking signal is more than 2, we are overestimating the forecast by using MA(4).

The analyst should check first the normality of the residuals before confirming the method instead of only deciding on the basis of convenience. Also, autocorrelation between lagged period must also be checked.

(c)

Alpha = 0.04 Beta = 0.16 Gamma = 0.27
Time(t) Sales A_t T_t S_t F(t) abs abs/y
1 4009 4009.0 0.0 1.0
2 4321 4022.3 2.2 1.0
3 4224 4032.9 3.6 1.0
4 3944 4032.6 2.9 1.0
5 4123 4039.2 3.5 1.0 4035.5 87.5 0.0212
6 4522 4059.4 6.3 1.0 4123.5 398.5 0.0881
7 4657 4088.3 10.0 1.0 4117.4 539.6 0.1159
8 4030 4096.4 9.7 1.0 4074.1 44.1 0.0109
9 4493 4121.4 12.2 1.0 4128.9 364.1 0.0810
10 4806 4153.4 15.4 1.1 4320.6 485.4 0.1010
11 4551 4176.4 16.7 1.1 4363.5 187.5 0.0412
12 4485 4207.2 19.0 1.0 4156.8 328.2 0.0732
13 4595 4236.5 20.7 1.0 4345.9 249.1 0.0542
14 4799 4266.0 22.1 1.1 4577.8 221.2 0.0461
15 4417 4283.2 21.3 1.1 4537.9 120.9 0.0274
16 4258 4300.5 20.7 1.0 4353.7 95.7 0.0225
17 4245 4508.9 263.9 0.0622
18 4900 4726.8 173.2 0.0353
19 4585 4585.0 0.0 0.0000
20 4533 4408.1 124.9 0.0275
MAPE for Test 0.0569
MAPE for Validation 0.0313

With the above set of alpha, beta, and gamma we were able to have lowest MAPE = 0.0313


Related Solutions

This data shows the series of quarterly shipments in millions of US dollars of US hosehold...
This data shows the series of quarterly shipments in millions of US dollars of US hosehold appliances between 1985 and 1989. Quarter Shipments Q1-1985 4009 Q1-1986 4123 Q1-1987 4493 Q1-1988 4595 Q1-1989 4245 Q2-1985 4321 Q2-1986 4522 Q2-1987 4806 Q2-1988 4799 Q2-1989 4900 Q3-1985 4224 Q3-1986 4657 Q3-1987 4551 Q3-1988 4417 Q3-1989 4585 Q4-1985 3944 Q4-1986 4030 Q4-1987 4485 Q4-1988 4258 Q4-1989 4533 (a) which of the following methods would be suitable for forecasting this series if applied to raw...
The following data shows the quarterly profit (in thousands of dollars) made by a particular company...
The following data shows the quarterly profit (in thousands of dollars) made by a particular company in the past 3 years. Year Quarter Profit ($1000s) 1 1 45 1 2 51 1 3 72 1 4 50 2 1 49 2 2 45 2 3 79 2 4 54 3 1 42 3 2 58 3 3 70 3 4 56 a.  Use α = 0.3 to compute the exponential smoothing values for the time series. Compute MSE and the forecast...
1.Assume you collected a quarterly sales data (in millions of dollars) over a four-year period from...
1.Assume you collected a quarterly sales data (in millions of dollars) over a four-year period from the first quarter 2016 to the fourth quarter 2019, and computed the seasonal index for each quarter. If the values of the actual sales and deseasonalised sales were 162 and 158.68, respectively, in the second quarter of 2016, what is the (normalised) seasonal index for the second quarter? Round your answer to two decimal places. 2. Assume you collected data representing the number of...
A consulting firm estimates the quarterly sales (in millions of dollars) of a Regional Airlines called...
A consulting firm estimates the quarterly sales (in millions of dollars) of a Regional Airlines called “U Fly”, by the following significant model: E(Sales) = 300.5 + 2.37(T) - 4.6Q1 - 6.8Q3 + 8.65Q4 Where E(Sales) is the average sales for each quarter (in millions of dollars), T is a time index defined as 1, 2, 3, …………,20, and 20 represents the first quarter of 2020, Q1 = 1 when in the first quarter, 0 otherwise, Q3 =1 when in...
The data below shows the annual salaries​ (in millions) and the number of viewers​ (in millions)...
The data below shows the annual salaries​ (in millions) and the number of viewers​ (in millions) of eight television actors and actresses. Answer parts ​a-c. Salary​ (x) 97 12 13 33 11 7 8 2 Viewers​ (y) 16 4.9 5.3 1.6 10.4 9.1 13.4 4.7 a. Find the value of the linear correlation coefficient r. b. Find the critical values of r from the table showing the critical values for the Pearson correlation coefficient using alpha=0.05. c. Is there sufficient...
Given the following data (in millions of dollars): contributedcapital in excess of par,    $70;...
Given the following data (in millions of dollars): contributed capital in excess of par,    $70; depreciation expense, $13; interest expense, $10; earnings after taxes, $32;    common stock ($1 par), $8; earnings before interest and taxes, $50; fixed assets, $73;    retained earnings, $82. The common stock price is $56. Compute the numerical    values of net fixed assets, earnings per share, price-earnings ratio, interest coverage    ratio, book value per share, and after-tax cash flow.
The following data represent a sample of the assets (in millions of dollars) of 28 credit...
The following data represent a sample of the assets (in millions of dollars) of 28 credit unions in a state. Assume that the population in this state is normally distributed with σ=3.5 million dollars. Use Excel to find the 99% confidence interval of the mean assets in millions of dollars. Round your answers to three decimal places and use ascending order. Assets 12.23 2.89 13.19 73.25 11.59 8.74 7.92 40.22 5.01 2.27 16.56 1.24 9.16 1.91 6.69 3.17 4.78 2.42...
12. The data below shows the annual salaries​ (in millions) and the number of viewers​ (in...
12. The data below shows the annual salaries​ (in millions) and the number of viewers​ (in millions) of eight television actors and actresses. Construct a​ scatterplot, find the value of the linear correlation coefficient​ r, and find the​ P-value using a=0.05. Is there sufficient evidence to conclude that there is a linear correlation between the two​ variables? Salary_(x)            Viewers_(y) 91           17 13           4.7 15           5.3 30           1.5 10           10.6 5              9.8 7              13.2 4              4.6 Construct the scatterplot. The linear...
The data below is the total spending (in millions of dollars) on drugs and other non-durable...
The data below is the total spending (in millions of dollars) on drugs and other non-durable products for your assigned state (or DC). You need to convert this data to spending per capita in constant 2019 dollars. Go to the FRED database at https://fred.stlouisfed.org/ Search for the PCEPI. Change the frequency to annual. Using that price index (this is a national index; there isn't a PCE index for each state), convert the following to 2019Q3 dollars. Again using the FRED...
Listed below are paired data consisting of amounts spent on advertising (in millions of dollars) and...
Listed below are paired data consisting of amounts spent on advertising (in millions of dollars) and the profits (in millions of dollars). Determine if there is a significant negative linear correlation between advertising cost and profit . Use a significance level of 0.01 and round all values to 4 decimal places. Advertising Cost Profit 3 18 4 22 5 16 6 29 7 24 8 31 9 22 10 29 11 25 Ho: ρ = 0 Ha: ρ < 0...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT