Provide Information Regarding Statistics & Econometrics : Standard Error of the Estimate & Coefficient of Determination Lecture 04

Read: Estimation of Regression Parameters

Goodness of Fit Test

The goodness of fit test is used to determine if the data adequately explain changes in the response variable. The standard error of the estimate and the coefficient of determination are two important goodness-of-fit measurements for regression analysis.
If the regression model successfully described an event or phenomenon, the actual and estimated values of response variable will be equal and the amount of deviation (called error) will be zero. As a result, the sum of squares deviation will be zero. In terms of regression line, this means that all observed data fall on the regression line and there is no variation or deviation of observed values from the regression line.
This ideal circumstance does not often exist in the field because in regression analysis, researchers and economist work with sample data rather than population data, and there is possibility of inaccuracy or error. This quantity of inaccuracy or error is measured by using an absolute measure of dispersion known as standard error of the estimate.
1. Standard Error of the Estimate
The standard error of the estimate is also called standard deviation of the regression and measure the average distance that the observed values fall from the regression line using the units of response variable. The population standard error of the estimate of Y on X (simple linear regression model) is denoted by and defined as:

The denominator (n – 2) is because two parameters are estimated in a simple linear regression model.

The standard error of the estimate is an absolute measure of the regression model, and it is difficult to comprehend. This absolute measure was enhanced by a relative measure known as the coefficient of determination denoted by R^2. The coefficient of determination is a relative measure of the regression model that determines how much variance in the response variable can be explained by the predictor variable.

2. Coefficient of Determination

Here, the total variation in the response variable is partitioned into explained variation (explained by the predictor) and unexplained variation (where the predictor fails to explain or variation is due to other predictors not included in the model).

The coefficient of determination is defined as the ratio of explained variation to total variation.

Practice Question 1.40

A survey of the pocket money of children in primary school was made by choosing at random four children of each of the ages 5, 7, 9 and 11 years. The amount of pocket money received is given below:

Age (years)	Pocket Money (Rs.)
5	2	8	10	12
7	9	13	14	16
9	9	14	16	21
11	18	19	23	36

Find the regression equation for predicting pocket money from age and also determine the standard error of the estimate and coefficient of determination.

Solution: let Pocket be denoted by Y and age be denoted by X. The OLS method is used to estimate the parameters of the model.

Y	X	XY	Squares of X	Squares of Y
2	5	10	25	4
8	5	40	25	64
10	5	50	25	100
12	5	60	25	144
9	7	63	49	81
13	7	91	49	169
14	7	98	49	196
16	7	112	49	256
9	9	81	81	81
14	9	126	81	196
16	9	144	81	256
21	9	189	81	441
18	11	198	121	324
19	11	209	121	361
23	11	253	121	529
36	11	396	121	1296
240	128	2120	1104	4498

The standard distance between the observations and the regression line is Rs 5 ages.

The variation in pocket money is 55.68% explained by age. In other words, 55.68% of the variation in pocket money is due to age.

Scatter Plot:

Using SPSS

Output

Project – I

Let's say we were curious to learn what consumers thought about online buying and wanted to research the variables that affect it. Let's focus on only four of them: i). convenience, ii). time-saving, iii). Web features, and iv). customers' level of trust.

These four elements are the foundation of a questionnaire that has been developed and circulated. The collected data are tabulated as:

No.	Questions	Strongly disagree	disagree	neutral	Agree	Strongly agree
	Convenience
1.	I receive my package on time by shopping online.	1	2	3	4	5
2.	When shopping online, detail information on item is offered.	1	2	3	4	5
3.	Online shopping allows me to buy products whenever I want every day.	1	2	3	4	5
4.	It is simple to select a product and evaluate it against competitors while shopping online	1	2	3	4	5
5	The need for travel is diminished by online buying.	1	2	3	4	5
6	Online shopping cuts down on energy use.	1	2	3	4	5
7	The home delivery is big advantage.	1	2	3	4	5
	Website Features
8	The layout of the website makes it simple for me to find things.	1	2	3	4	5
9	When buying online, the website's design facilitates my search for and selection of the appropriate item.	1	2	3	4	5
10	The product review may be viewed here.	1	2	3	4	5
11	The return or change policy of product is specified.	1	2	3	4	5
12	Addressing inquiries about the product to making a purchase.	1	2	3	4	5
13	The chatbot available 24/7	1	2	3	4	5
14	The website is available on mobile app.	1	2	3	4	5
15	The website will be multilingual support.	1	2	3	4	5
16	I always discovered things advertised online in accordance with market norms.	1	2	3	4	5
	Time Saving
17.	Product selection and evaluation require less time.	1	2	3	4	5
18.	Time is saved by shopping online.	1	2	3	4	5
19.	Product selection and evaluation require less time.	1	2	3	4	5
20.	Online shopping provides the opportunity to use potential time for potential work.	1	2	3	4	5
21.	Online shopping makes it simple to send gifts to a loved one in time for upcoming events.	1	2	3	4	5
	Trust
22.	Your expectation is met by the online purchasing process.	1	2	3	4	5
23.	Online transactions are secure.	1	2	3	4	5
23.	In the context of Pakistan, online purchasing is quite practical or efficient.	1	2	3	4	5
24.	loyal customers can be attracted through online shopping.	1	2	3	4	5
25.	Online retailers charge fair pricing for their goods	1	2	3	4	5
26.	The websites are comfortable in searching.	1	2	3	4	5