Question

a. What is the Root Mean Squared Error (RMSE) of forecasts? Write out the formula. What is the relevance of the concept in forecasting? b. What is the residual standard error of regression/SER (that is generally reported with regression results - Write out the formula. c. How is RMSE similar to SER? How is it different? Which of the two concepts are more relevant for forecasting? Why?

Answer 1

KoAns ) Forecasting : Forecasting is a technique that uses historical data as inputs to make informed estimates that are predictive in determining the direction of future trends. Businesses utilize forecasting to determine how to allocate their budgets or plan for anticipated expenses for an upcoming period of time. This is typically based on the projected demand for the goods and services ofoffered.

Relevance of forecasting

1. Promotion of new business:

Forecasting is of utmost importance in setting up a new business. It is not an easy task to start a new business as it is full of uncertainties and risks. With the help of forecasting the promoter can find out whether he can succeed in the new business; whether he can face the existing competition; what is the possibility of creating demand for the proposed product etc.

2. Estimation of financial requirements:

The importance of forecasting can’t be ignored in estimating the financial requirements of a concern. Efficient utilisation of capital is a delicate issue before the management. No business can survive without adequate capital. But adequacy of either fixed or working capital depends entirely on sound financial forecasting.

Financial estimates can be calculated in the light of probable sales and cost thereof. How much capital is needed for expansion, development etc.

3. Smooth and continuous working of a concernForecasting of earnings’ ensures smooth and continuous working of an enterprise, particularly to newly established ones. By forecasting, these concerns can estimate their expected profits or losses. The object of a forecast is to reduce in black and white the details of working of a concern.

4. Correctness of management decisions:

The correctness of management decisions to a great extent depends upon accurate forecasting. As Meivin, T. Copeland says, “Administration is essentially a decision making process and authority has responsibility for making decisions and for ascertaining that the decisions made are carried out.

In business, whether the enterprise is large or small, changes in conditions occur; shifts in personnel take place, unforeseen contingencies arise. Moreover, just to get the wheels started and to keep them turning, decisions must be made.”

Root Mean Square Error (RMSE)

Root Mean Square Error (RMSE) is the standard deviation of the residuals (prediction errors). Residuals are a measure of how far from the regression line data points are; RMSE is a measure of how spread out these residuals are. In other words, it tells you how concentrated the data is around the line of best fit. Root mean square error is commonly used in climatology, forecasting, and regression analysis to verify experimental results

The formula is:

Where:

• f = forecasts (expected values or unknown results),
• o = observed values (known results).

The bar above the squared differences is the mean (similar to x̄). The same formula can be written with the following, slightly different, notation (Barnston, 1992):

Where:

• Σ = summation (“add up”)
• (zfi – Zoi)Sup>2 = differences, squared
• N = sample size.

You can use whichever formula you feel most comfortable with, as they both do the same thing. If you don’t like formulas, you can find the RMSE by:

1. Squaring the residuals.
2. Finding the average of the residuals.
3. Taking the square root of the result.

That said, this can be a lot of calculation, depending on how large your data set it. A shortcut to finding the root mean square error is:

Where SDy is the standard deviation of Y.

When standardized observations and forecasts are used as RMSE inputs, there is a direct relationship with the correlation coefficient. For example, if the correlation coefficient is 1, the RMSE will be 0, because all of the points lie on the regression line (and therefore there are no errors).

Ans B ) Standard error of regression

The residual standard deviation is a statistical term used to describe the difference in standard deviations of observed values versus predicted values as shown by points in a regression analysis. Regression analysis is a method used in statistics to show a relationship between two different variables, and to describe how well you can predict the behavior of one variable from the behavior of another.

Residual standard deviation is also referred to as the standard deviation of points around a fitted line or the standard error of estimate.

The Formulas for Residual and Residual Standard Deviation Is

The Formulas for Residual and Residual Standard Deviation Is

=. ( Y - Yest )

= √ ( Y - Yest )2 / n - 2

Sres = residual s.d.

Y = observed value

Yest = estimated value

N = data points

Ans C ) Difference between RMSE and SER

Standard deviation is used to measure the spread of data around the mean, while RMSE is used to measure distance between some values and prediction for those values. RMSE is generally used to measure the error of prediction, i.e. how much the predictions you made differ from the predicted data.

both standard deviation and RMSE are similar because they are square roots of squared differences between some values. Nonetheless, they are not the same. Standard deviation is used to measure the spread of data around the mean, while RMSE is used to measure distance between some values and prediction for those values. RMSE is generally used to measure the error of prediction, i.e. how much the predictions you made differ from the predicted data. If you use mean as your prediction for all the cases, then RMSE and SD will be exactly the same.

As a sidenote, you may notice that mean is a value that minimizes the squared distance to all the values in the sample. This is the reason why we use standard deviation along with it -- they are related

The only difference is that you divide by nn and not n−1n−1 since you are not subtracting the sample mean here. The RMSE would then correspond to σσ . Therefore, the population RMSE is σσ and you want a CI for that.