As you can see, there is exactly one straight line that passes through the two data points. Note that we must distinguish carefully between the unknown parameters that we denote by capital letters and our estimates of them, which we denote by lower-case letters.
Many textbooks use Greek letters for the former and corresponding Roman letters for the latter. This equation is a "perfect fit" for the data, in the sense that both data points lie exactly on the line.
In the case of only two data points, choosing the best straight line to represent the relationship defined by the data is easy! The slope of this line is b 1 , which is our empirical estimate of B 1 , while the value of E where the best-fit line intercepts the vertical axis is b 0 , our estimate of B 0.
As discussed in the page on economic models, we can rarely expect the relationship between two economic variables to be "perfect.
Differences in these other variables between observations will cause some data points to lie above the regression line and others to lie below it. Figure 2 shows a sample of three data points. No single line passes through all three points. Choosing the line passing through any two of the three points leaves one point off the line, so we say that there is one degree of freedom in choosing the line.
In the case of only two points, there were zero degrees of freedom; if we added a fourth point, there would be two degrees of freedom. Many alternative criteria could be chosen for picking a "best-fit" line for three or more data points. The most common methods involve trying to make the residuals , the deviations of the data points from the estimated regression line, as small as possible.
In the case of only two data points, our regression line passes through both points, so the residuals are zero--the data points do not deviate from the line. With three or more data points we cannot find a line that makes all the residuals zero, except in the unusual case where all the points happen to lie on the same line.
By far the most common estimator is the least-squares regression line. This is the line that makes the sum of the squared residuals as small as possible, where the residual is measured as the vertical distance of the point from the estimated regression line. The required number of sample points does depend on objects. If you are doing exploratory analysis just to see if one model say linear in a covariate looks better than another say a quadratic function of the covariate less than 10 points may be enough.
But if you want very accurate estimates of the correlation and regression coefficients for the covariates you could need more than 10 per covariate. An accuracy of prediction criterion could require even more samples than accurate parameter estimates. Note that the variance of the estimates and prediction all involve the variance of the models error term. Sign up to join this community. The best answers are voted up and rise to the top.
Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Minimal number of points for a linear regression Ask Question. Asked 9 years, 1 month ago. Active 1 month ago. Viewed 51k times.
Improve this question. Peter Flom Francoise Francoise 1 1 gold badge 2 2 silver badges 3 3 bronze badges. If they include estimates of variability, then two could be enough using a t-test or its analog. The basic statistical principle that applies here is that when random variation is an unlikely explanation of what you are observing, then you have the right to attribute any apparent trend to non-random causes. When the trend is strong, very few data values may be needed to come to such a conclusion, all generic "rules of thumb" notwithstanding.
I was wondering whether you would have any advice on building a regression model on a very small datasets. But I only have 27 data points, 27 years of annual data. Any advice would be much appreciated. Do your best, making as many assumptions as you need. Get more data. Or could each business be a data point?
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