What do r squared values tell you

Essentially, R-squared is a statistical analysis technique for the practical use and trustworthiness of betas of securities. R-squared will give you an estimate of the relationship between movements of a dependent variable based on an independent variable's movements. It doesn't tell you whether your chosen model is good or bad, nor will it tell you whether the data and predictions are biased.

A high or low R-square isn't necessarily good or bad, as it doesn't convey the reliability of the model, nor whether you've chosen the right regression. You can get a low R-squared for a good model, or a high R-square for a poorly fitted model, and vice versa. In some fields, such as the social sciences, even a relatively low R-Squared such as 0. In other fields, the standards for a good R-Squared reading can be much higher, such as 0.

In finance, an R-Squared above 0. This is not a hard rule, however, and will depend on the specific analysis. Essentially, an R-Squared value of 0. For instance, if a mutual fund has an R-Squared value of 0. Here again, it depends on the context. Suppose you are searching for an index fund that will track a specific index as closely as possible.

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Develop and improve products. List of Partners vendors. That is, create a plot of the observed data and the predicted values of the data. This can reveal situations where R-Squared is highly misleading. For example, if the observed and predicted values do not appear as a cloud formed around a straight line, then the R-Squared , and the model itself, will be misleading.

Similarly, outliers can make the R-Squared statistic be exaggerated or be much smaller than is appropriate to describe the overall pattern in the data.

In 25 years of building models, of everything from retail IPOs through to drug testing, I have never seen a good model with an R-Squared of more than 0. Such high values always mean that something is wrong, usually seriously wrong.

What is a good R-Squared value? Well, you need to take context into account. There are a lot of different factors that can cause the value to be high or low.

This makes it dangerous to conclude that a model is good or bad based solely on the value of R-Squared. For example:. For the R-Squared to have any meaning at all in the vast majority of applications it is important that the model says something useful about causality. Consider, for example, a model that predicts adults' height based on their weight and gets an R-Squared of 0. Is such a model meaningful? It depends on the context. But, for most contexts the model is unlikely to be useful.

The implication, that if we get adults to eat more they will get taller, is rarely true. But, consider a model that predicts tomorrow's exchange rate and has an R-Squared of 0.

If the model is sensible in terms of its causal assumptions, then there is a good chance that this model is accurate enough to make its owner very rich. A natural thing to do is to compare models based on their R-Squared statistics. If one model has a higher R-Squared value , surely it is better? This is, as a pretty general rule, an awful idea. There are two different reasons for this:.

Technically, R-Squared is only valid for linear models with numeric data. While I find it useful for lots of other types of models, it is rare to see it reported for models using categorical outcome variables e. Many pseudo R-squared models have been developed for such purposes e.

These are designed to mimic R-Squared in that 0 means a bad model and 1 means a great model. However, they are fundamentally different from R-Squared in that they do not indicate the variance explained by a model. No such interpretation is possible. In particular, many of these statistics can never ever get to a value of 1.

Market research Social research commercial Customer feedback Academic research Polling Employee research I don't have survey data. There is no seasonality in the income data. In fact, there is almost no pattern in it at all except for a trend that increased slightly in the earlier years.

This is not a good sign if we hope to get forecasts that have any specificity. By comparison, the seasonal pattern is the most striking feature in the auto sales, so the first thing that needs to be done is to seasonally adjust the latter. Seasonally adjusted auto sales independently obtained from the same government source and personal income line up like this when plotted on the same graph:.

The strong and generally similar-looking trends suggest that we will get a very high value of R-squared if we regress sales on income, and indeed we do. Here is the summary table for that regression:. However, a result like this is to be expected when regressing a strongly trended series on any other strongly trended series , regardless of whether they are logically related.

Here are the line fit plot and residuals-vs-time plot for the model:. The residual-vs-time plot indicates that the model has some terrible problems. First, there is very strong positive autocorrelation in the errors, i.

In fact, the lag-1 autocorrelation is 0. It is clear why this happens: the two curves do not have exactly the same shape. The trend in the auto sales series tends to vary over time while the trend in income is much more consistent, so the two variales get out-of-synch with each other. This is typical of nonstationary time series data. And finally, the local variance of the errors increases steadily over time.

The reason for this is that random variations in auto sales like most other measures of macroeconomic activity tend to be consistent over time in percentage terms rather than absolute terms, and the absolute level of the series has risen dramatically due to a combination of inflationary growth and real growth. As the level as grown, the variance of the random fluctuations has grown with it.

Confidence intervals for forecasts in the near future will therefore be way too narrow, being based on average error sizes over the whole history of the series. So, despite the high value of R-squared, this is a very bad model. One way to try to improve the model would be to deflate both series first. This would at least eliminate the inflationary component of growth, which hopefully will make the variance of the errors more consistent over time.

Here is a time series plot showing auto sales and personal income after they have been deflated by dividing them by the U. This does indeed flatten out the trend somewhat, and it also brings out some fine detail in the month-to-month variations that was not so apparent on the original plot. In particular, we begin to see some small bumps and wiggles in the income data that roughly line up with larger bumps and wiggles in the auto sales data.

If we fit a simple regression model to these two variables, the following results are obtained:. Adjusted R-squared is only 0. Well, no. Because the dependent variables are not the same, it is not appropriate to do a head-to-head comparison of R-squared. Arguably this is a better model, because it separates out the real growth in sales from the inflationary growth, and also because the errors have a more consistent variance over time.

The latter issue is not the bottom line, but it is a step in the direction of fixing the model assumptions. Most interestingly, the deflated income data shows some fine detail that matches up with similar patterns in the sales data. However, the error variance is still a long way from being constant over the full two-and-a-half decades, and the problems of badly autocorrelated errors and a particularly bad fit to the most recent data have not been solved.

Another statistic that we might be tempted to compare between these two models is the standard error of the regression, which normally is the best bottom-line statistic to focus on. But wait… these two numbers cannot be directly compared, either, because they are not measured in the same units.

The standard error of the first model is measured in units of current dollar s, while the standard error of the second model is measured in units of dollar s. Those were decades of high inflation, and dollars were not worth nearly as much as dollars were worth in the earlier years. In fact, a dollar was only worth about one-quarter of a dollar. The slope coefficients in the two models are also of interest. Because the units of the dependent and independent variables are the same in each model current dollars in the first model, dollars in the second model , the slope coefficient can be interpreted as the predicted increase in dollars spent on autos per dollar of increase in income.

The slope coefficients in the two models are nearly identical: 0. Notice that we are now 3 levels deep in data transformations: seasonal adjustment, deflation, and differencing! This sort of situation is very common in time series analysis. This model merely predicts that each monthly difference will be the same, i.

Adjusted R-squared has dropped to zero! We should look instead at the standard error of the regression. The units and sample of the dependent variable are the same for this model as for the previous one, so their regression standard errors can be legitimately compared.

The sample size for the second model is actually 1 less than that of the first model due to the lack of period-zero value for computing a period-1 difference, but this is insignificant in such a large data set. The regression standard error of this model is only 2. The residual-vs-time plot for this model and the previous one have the same vertical scaling: look at them both and compare the size of the errors, particularly those that have occurred recently.

It is often the case that the best information about where a time series is going to go next is where it has been lately. There is no line fit plot for this model, because there is no independent variable, but here is the residual-versus-time plot:. These residuals look quite random to the naked eye, but they actually exhibit negative autocorrelation , i. The lag-1 autocorrelation here is