Question

Year	Quarter	SPeClty
2006	3	4480.77
	4	5827.48
2007	1	5602.84
	2	5269.74
	3	4945.67
	4	5124.22
2008	1	4996.31
	2	5147.61
	3	4903.32
	4	4960.84
2009	1	4864.16
	2	4785.21
	3	4684.43
	4	4732.58
2010	1	4947.63
	2	4826.21
	3	4801.46
	4	4789.05
2011	1	4812.69
	2	4787.65
	3	4871.31
	4	4756.61
2012	1	4583.15
	2	4659.09
	3	4587.55
	4	4746.01
2013	1	4556.63
	2	4632.57
	3	4561.03
	4	4719.49
2014	1	4516.85
	2	4592.78
	3	4521.24
	4	4844.6
2015	1	4588.54
	2	4571.84
	3	4671.24
	4	5011.6
2016	1	4821.54
	2	4916.84
	3	4791.31
	4	5183.49
2017	1	5045.38
	2	5138.97
	3	4915.37
	4	5369.34
2018	1	5379.83
	2	5481
	3	5265.31
	4	5579.14
2019	1	5743.98
	2	5750.01
	3	5542.86

a) in a paragraph write about Trend and Seasonality of this store.

b) write about Regression model(s) and examine hypotheses

c) what is the least square regression line?

d) write about standard error and interpret it in a short paragraph.

* please just do it with excel and explain how you are solving it (preferred to use excel screenshots)

Answer 1

Given:

Assuming the column names as a year, quarter and sales. The data given is the time series of sales of a particular store from the third quarter of 2006 till the third quarter of 2019.

To find:

a) Short note on Trend and Seasonality of this store
b) Regression model(s) and examine hypotheses
c) The least-square regression line
d) The standard error and its interpretation

Solution:

a) Short note on Trend and Seasonality of this store

Trend: smooth, regular long term movement of time series

Seasonality: periodic fluctuations in a time series

The plot of sales is as follows:

We see that there is no trend and no seasonality present. However, note the sales first decrease and then gradually increase

b) Regression model(s) and examine hypotheses

The regression models are

1. Autoregressive model:

The model, for lag p, is given by

where xt is the value at time t and xt-p is the value at time t-p.

Precisely, for lag =1, ie AR(1). We have

That is, we regress the current value on the previous value

2. Moving average models:

The model for lags = 1 is ie MA(1)

where c0 is a constant and at and at-1 are white noise series.

We can have lag = q

3. Autoregressive moving average model:

this model is a combination of autoregressive and moving average model.

The ARMA(1,1) model is given by

For all the three models we test for the significance of the coefficients using t-test.

c) The least-square regression line:

The least-square regression line is obtained from the least square estimates.

The least-square regression line is the fit line obtained by the method of ordinary least squares. The estimates obtained are unbiased and are such that they minimize the error.

d) The standard error and its interpretation:

The standard error is the difference between the actual value and the predicted value. Lower the standard error better the predicted value. The sum of standard errors obtained from the least square method is zero.