Question 1
Construct a line chart for your given brand of rice. Comment on any features of interest.
Question 2
Estimate a model of volume sales using a trend component. In other words, estimate the following:
Where yt = Volume sales in period t
T = Trend
Comment on your findings. How do your findings for the trend variable compare to
your observations in step 1?
Hint : To complete this step, you must create a trend variable.
Question 3
Refine the model you created in step 2 by adding a seasonal index. In other words, estimate the following:
Where SI = Seasonal Index
Are there any months that exhibit above average sales? What do you think is the reason? How do your findings compare to your observations in step 1? Comment on your findings.
Hint : To complete this step, you must create a seasonal index.
Question 4
In the mid 2005, the outbreak of a new disease Rhizoctonia Solani had a devastating impact on the rice farming industry all over the world. Known as ‘Sheath Blight’, fear of the disease quickly spread throughout India causing people to avoid consumption of certain brands of rice. Many consumers seek alternative food such as bread as a substitute for the 1 year of the plague. At the same time, sales of certain brands of rice were not affected due to the country of production.
Sales Volume of Rice Brand C
One of the major responsibilities of the AgriFood and Veterinary Authority of India is to supervise and coordinate the production of all the imported agricultural productions of India. Thus, the director of the company is interested to identify and analyse the issues that are related to the economy of rice. Nine major rice brands have been considered for the study. The brands of rice are stored in stores of New Delhi and Mumbai. Nine analysing groups were appointed to analyse nine different rice brands. This group was appointed to analyse the various aspects of rice brand C. The necessary analyses are given in details in the following sections.
There are several time series components. Any time series data always follows any one of the four different types of time series components. The four time series components are discussed as below:
 Secular Trend: This is the main time series component and it shows the effects of the socioeconomic and political factors in the long term. This may be increasing and decreasing trend (Goldthorpe 2016).
 Seasonal trend: This type of trend shows short term changes in the trend. For example, the sale of winter garments is high in winter than in summer. This trend is repeated over years (Piskorski et al. 2018).
 Cyclic Movements: The long term oscillations that occur in a time series is known as cyclic movements. In most of the economic data, this type of trend is visible (Rey 2015).
 Irregular Fluctuations:When there are sudden fluctuations in a data, it is known as irregular fluctuations. These types of fluctuations are usually not expected to be repeated (Costa and Goldberger 2015).
The volume of tonnes of rice (in thousands) consumed is shown with the help of the following line chart. It can be seen from the chart very clearly that the volume of consumption of the chosen brand of rice has shown great fluctuations over the time. There has been observed an increased trend in the consumption volume of rice of Mumbai C. The volume of consumption has increased in some years and has decreased in some other years but overall, there is an increased trend in the consumption volume. This type of trend is seasonal trend.
The volume of sales can be estimated from the given volume over the time by the following equation, which has been obtained as a result of the regression analysis:
Estimated sales volume (y_{t}) = 5.9644 + 0.0044 * Trend (T)
From the computed results, it can be said that in the absence of trend, the estimated sales volume will be 5.964 tonnes. With each unit increase in the trend value, the estimated sales volume increases by 0.004 times. Thus, the increase in the sales volume is very little over the years. This has also been observed from the discussion in part 1 that the volume of consumption increases at a very little rate.
Again, it can be said from the regression analysis that the value of R Square is 0.5151, which indicates that 51.51 percent of the variations in the sales volume can be explained by the trend. From the significance value obtained in table 2.2, it can be said that the model is significant as the significance value is less than the 95 percent level of significance (0.05). The table showing the outputs of the regression analysis is given below:
Table 2.1: Regression Statistics 

Multiple R 
0.717711731 
R Square 
0.51511013 
Adjusted R Square 
0.51317057 
Standard Error 
0.311014991 
Observations 
252 
Table 2.2: ANOVA 


df 
SS 
MS 
F 
Significance F 
Regression 
1 
25.68973554 
25.68973554 
265.5809911 
3.55113E41 
Residual 
250 
24.18258121 
0.096730325 

Total 
251 
49.87231675 
Regression Analysis of Sales Volume
Table 2.3: Regression Coefficients

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
5.964398827 
0.039301116 
151.7615653 
3.7224E248 
5.886995345 
6.041802309 
Trend 
0.004389075 
0.000269324 
16.29665583 
3.55113E41 
0.003858643 
0.004919508 
With the addition of seasonal index, the model developed in question 2 has been refined as follows:
Here SI is the seasonal index. The seasonal indices have been calculated and from the results it can be seen that the seasonal indices for the months of April, May, July and September has been greater than 100. Thus it can be said that these four months exhibit sales of rice above the average sales. The months of April, May, July and September are usually summer and monsoon seasons in India. The production of rice in these seasons are high as rice grows better in this season. As the production of rice is high, the consumption of rice in these seasons are also high. Thus, the sales in these seasons are higher than the average sales. The results of the regression analysis for the revised model is given the following tables 3.1, 3.2 and 3.3. From the R Square value obtained from table 3.1, it can be said that the revised model so developed with the introduction of the seasonal indices can explain 67.53 percent of the variability in the sales of rice of brand C.
Table 3.1: Regression Statistics 

Multiple R 
0.821814993 
R Square 
0.675379882 
Adjusted R Square 
0.672772491 
Standard Error 
0.254986868 
Observations 
252 
Table 3.2: ANOVA 


df 
SS 
MS 
F 
Significance F 
Regression 
2 
33.6827594 
16.8413797 
259.0252132 
1.46636E61 
Residual 
249 
16.18955735 
0.065018303 

Total 
251 
49.87231675 
Table 3.3: Regression Coefficients

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0.349374969 
0.570355201 
0.612556821 
0.540728426 
1.472710534 
0.773960596 
Trend 
0.004350414 
0.000220834 
19.69996125 
1.03249E52 
0.003915474 
0.004785354 
Seasonal Index 
0.063186645 
0.005698854 
11.08760488 
1.75403E23 
0.051962542 
0.074410748 
Table 3.4: Seasonal Indices of the months of the Disease
Month 
Seasonal Index 
Jun 
99.90 
Jul 
100.77 
Aug 
99.14 
Sep 
100.78 
Oct 
99.14 
Nov 
98.43 
Dec 
97.63 
Jan 
97.59 
Feb 
97.98 
Mar 
98.73 
Apr 
101.48 
May 
108.42 
In the mid 2005, a new disease named Rhizoctonia Solani came into effect and had a devastating effect on the industry that farms rice all over the world. The fear of this disease known as the ‘Sheath Blight’ spread very quickly in India and this made a lot of people avoid rice consumption of certain brands. People had switched to an alternative form of food such as bread for that one year of the plague. Certain rice brand did not have any effect in the sales as India was the country where rice was produced. Now, the effect of this disease has been considered in the model devised earlier and developed. A dummy variable named disease has been introduced which takes the values 0 and 1 where 0 indicates the absence of the disease and 1 indicates the presence of the disease. The developed and revised model is stated as follows.
From the obtained equation, it can be said that there is a positive effect of the disease on the sales of the selected brand of Rice (Mumbai C). In the presence of the disease the sales of rice increases by 0.7488. In the absence of the disease, there is no effect in the estimated sales. The results of the regression analysis of the model revised with the introduction of another variable disease is given in tables 3.1, 3.2 and 3.3. The r square value of the revised model with the introduction of the new variable stating the presence or absence of the disease is 0.8039 (Table 4.1). This indicates that 80.39 percent of the variability in the sales of rice of brand C can be explained with the help of this model. Thus this model can be termed as a model better than the ones developed previously. From the significance value also given in table 4.3 it can be said that the model so estimated is quite significant.
Table 4.1: Regression Statistics 

Multiple R 
0.896580777 
R Square 
0.803857089 
Adjusted R Square 
0.801484393 
Standard Error 
0.198605028 
Observations 
252 
Table 4.2: ANOVA 


df 
SS 
MS 
F 
Significance F 
Regression 
3 
40.09021537 
13.36340512 
338.7947378 
2.15027E87 
Residual 
248 
9.782101376 
0.039443957 

Total 
251 
49.87231675 
Revised Model with Seasonal Indices
Table 4.3: Regression Coefficients

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0.385030553 
0.444248984 
0.866699906 
0.386944212 
1.260012511 
0.489951405 
Trend 
0.004350414 
0.000172004 
25.29256917 
1.26152E70 
0.00401164 
0.004689188 
Seasonal Index 
0.063186645 
0.004438743 
14.23525711 
5.15873E34 
0.054444206 
0.071929085 
Disease 
0.748767252 
0.05874816 
12.74537377 
5.9073E29 
0.633058311 
0.864476194 
Price elasticity of demand captures proportionate change in quantity demanded in response to proportionate change in price (Labandeira, Labeaga and LópezOtero 2017). It measures responsiveness of demand for a corresponding price change. Demand changes in response to both own price of the product and price of the related product. When elasticity is computed for capturing demand response for own price, then it is called own price elasticity. Because of the inverse relation between price and demand, own price elasticity has a negative sign. Demand response for change in price of the related product is computed from the cross price elasticity of demand. The sign of cross price elasticity of demand depends on the nature of the relation. For substitute products, cross price elasticity is positive while for complementary goods it is negative.
Taking into account the effects of trend, Seasonal Indices, disease, own price and competitor price, the following relationship can be established:
Here, is the own price of rice in the period t and is the B^{th} competitor price of the good in period t. s
It can be said from the analysis that the price of Mumbai B brand of rice has been observed as the only significant brand of rice which has an effect on the sales of Mumbai C brand of rice. All the other brands of rice that are available does not have any significant relationship on the sales of Mumbai C brand of rice.
From the obtained relationship, it can be said that the sales of Mumbai C brand of rice increases if the price of that brand falls and if the price of the competitive brand B increases. Thus, the price of brand C must always be kept less than the price of brand B in order to increase the sales.
Table 5.1: Regression Statistics considering price effects of all rice brands 

Multiple R 
0.910142885 
R Square 
0.828360071 
Adjusted R Square 
0.819742166 
Standard Error 
0.189251785 
Observations 
252 
Table 5.2: ANOVA considering price effects of all rice brands 

df 
SS 
MS 
F 
Significance F 

Regression 
12 
41.31223582 
3.442686318 
96.12082374 
3.11232E84 
Residual 
239 
8.560080927 
0.035816238 

Total 
251 
49.87231675 
Table 5.3: Regression Coefficients considering price effects of all rice brands and identifying insignificant Price Effects
Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Remarks 

Intercept 
1.005 
0.559 
1.800 
0.073 
2.106 
0.095 

Trend 
0.004 
0.000 
25.685 
0.000 
0.004 
0.005 

Seasonal Index 
0.064 
0.004 
14.782 
0.000 
0.055 
0.072 

Disease 
0.737 
0.059 
12.514 
0.000 
0.621 
0.853 

Price C 
0.002 
0.000 
4.193 
0.000 
0.003 
0.001 

Price A 
0.001 
0.000 
1.647 
0.101 
0.000 
0.002 
drop 
Price B 
0.001 
0.000 
3.105 
0.002 
0.000 
0.002 

Price D 
0.000 
0.000 
0.810 
0.418 
0.001 
0.001 
drop 
Price E 
0.000 
0.000 
1.653 
0.100 
0.000 
0.001 
drop 
Price F 
0.000 
0.000 
0.610 
0.542 
0.001 
0.001 
drop 
Price G 
0.000 
0.000 
0.156 
0.876 
0.000 
0.000 
drop 
Price H 
0.000 
0.000 
0.374 
0.709 
0.001 
0.000 
drop 
Price I 
0.000 
0.000 
1.302 
0.194 
0.000 
0.001 
drop 
Table 5.4: Regression Statistics dropping out insignificant price effects 

Multiple R 
0.90727541 
R Square 
0.823148669 
Adjusted R Square 
0.81955413 
Standard Error 
0.189350469 
Observations 
252 
Table 5.5: ANOVA dropping out insignificant price effects 


df 
SS 
MS 
F 
Significance F 
Regression 
5 
41.05233115 
8.21046623 
228.9997723 
2.22832E90 
Residual 
246 
8.819985594 
0.0358536 

Total 
251 
49.87231675 
Table 5.6: Regression coefficients dropping out insignificant price effects

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0.280 
0.441 
0.636 
0.526 
1.148 
0.588 
Trend 
0.004 
0.000 
25.949 
0.000 
0.004 
0.005 
Seasonal Index 
0.063 
0.004 
14.903 
0.000 
0.055 
0.072 
Disease 
0.746 
0.057 
13.106 
0.000 
0.634 
0.859 
Price C 
0.002 
0.000 
3.942 
0.000 
0.002 
0.001 
Price B 
0.001 
0.000 
2.968 
0.003 
0.000 
0.002 
The Chow test has been developed by The econometrician Gregory Chow in 1960. This test is mainly used to test whether the coefficients of regression analysis on two different datasets are equal. This type of test is mostly used in case of time series data where the structural break present in a trend is tested. Thus it can be said that whether the impact of the independent variable on different subgroups of a dataset are same or different is evaluated with the help of Chow Test (Asteriou and Hall 2015).
Model with the Presence of a Disease
Mathematically, If the regression model is , then the regression model can be broken into two periods as and . The null hypothesis for this test can be given by,
Null hypothesis:
The test statistic for the Chow Test can be given by: (Nielsen and Whitby 2018).
Here, denotes the residual sum of squares for the full model, denotes the residual sum of squares for the first part of the model and denote the residual sum of squares for the second part of the model, is the number of observations in first part, is the number of observations in second part and p = 3.
It can be seen that the test statistic follows an F distribution with and degrees of freedom.
For the chow test, the data that has been considered for regression has been divided into two parts, the first group contains data from June 1995 to November 2005 and the second group contains data from December 2005 to May 2016. Regression were run separately on these two groups of data for the Chow test. From the analysis, the value of the test statistic has been obtained as 4.605 which is greater than the critical value of F at 95 percent level of significance (2.641). This indicates that the null hypothesis is rejected. The regression coefficients of the two separate groups are not equal. Thus, the effect of the independent variables on the dependent variable for both the groups are not same.
Table 6.1: Regression Statistics Group 1 

Multiple R 
0.854166 
R Square 
0.729599 
Adjusted R Square 
0.718333 
Standard Error 
0.186277 
Observations 
126 
Table 6.2: ANOVA Group 1 


df 
SS 
MS 
F 
Significance F 
Regression 
5 
11.23504 
2.247008 
64.75717 
1.89E32 
Residual 
120 
4.163879 
0.034699 

Total 
125 
15.39892 
Table 6.3: Coefficients of Regression Analysis of Group 1

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0.859 
0.648 
1.325 
0.188 
2.143 
0.424 
Trend 
0.003 
0.000 
6.990 
0.000 
0.002 
0.004 
Seasonal Index 
0.069 
0.006 
11.337 
0.000 
0.057 
0.081 
Disease 
0.657 
0.086 
7.674 
0.000 
0.487 
0.826 
Price C 
0.002 
0.001 
3.269 
0.001 
0.003 
0.001 
B 
0.002 
0.001 
3.017 
0.003 
0.001 
0.003 
Table 6.4: Regression Statistics Group 2 

Multiple R 
0.840 
R Square 
0.706 
Adjusted R Square 
0.693 
Standard Error 
0.187 
Observations 
126 
Table 6.5: ANOVA Group 2 

df 
SS 
MS 
F 
Significance F 

Regression 
5 
10.041 
2.008 
57.551 
0.000 
Residual 
120 
4.187 
0.035 

Total 
125 
14.228 
Table 6.6: Coefficients of Regression Analysis of Group 2
Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 

Intercept 
0.071 
0.602 
0.119 
0.906 
1.121 
1.264 
Trend 
0.005 
0.000 
9.532 
0.000 
0.004 
0.006 
Seasonal Index 
0.059 
0.006 
9.892 
0.000 
0.047 
0.071 
Disease 
0.916 
0.087 
10.538 
0.000 
0.744 
1.088 
Price C 
0.001 
0.001 
2.420 
0.017 
0.003 
0.000 
B 
0.001 
0.001 
1.794 
0.075 
0.000 
0.002 
Table 6.7: Results of Chow Test
Chow test Statistic 
4.605178 
Critical F 
2.641296 
References
Costa, M.D. and Goldberger, A.L., 2015. Generalized multiscale entropy analysis: application to quantifying the complex volatility of human heartbeat time series. Entropy, 17(3), pp.11971203. Available at: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4834981/
Goldthorpe, J.H., 2016. Social class mobility in modern Britain: Changing structure, constant process. Journal of the British Academy, 4, pp.89111. Available at: https://www.britac.ac.uk/sites/default/files/05%20Goldthorpe%201825.pdf
Labandeira, X., Labeaga, J.M. and LópezOtero, X., 2017. A metaanalysis on the price elasticity of energy demand. Energy Policy, 102, pp.549568. Available at: https://www.pubmanitoba.ca/v1/proceedingsdecisions/applcurrent/pubs/2017%20mh%20gra/irs%20to%20iec/mhdaymark%20%28load%29%20i12%20attachment3%20%20labandeira.pdf.
Nielsen, B. and Whitby, A. (2018). A Joint Chow Test for Structural Instability, available at: https://www.mdpi.com/22251146/3/1/156.
Piskorski, J., Kosmider, M., Mieszkowski, D., Krauze, T., Wykretowicz, A. and Guzik, P., 2018. Properties of Asymmetric Detrended Fluctuation Analysis in the time series of RR intervals. Physica A: Statistical Mechanics and its Applications, 491, pp.347360. Available at: https://www.sciencedirect.com/journal/physicaastatisticalmechanicsanditsapplications/vol/491
Rey, H., 2015. Dilemma not trilemma: the global financial cycle and monetary policy independence (No. w21162). National Bureau of Economic Research. Available at: https://www.nber.org/papers/w21162
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