![]() We dealt with the first of these in ordinary linear regression (no log transformation). In the case where there is one independent variable x, there are four ways of making a log transformation, namely ![]() Thus the equivalent of the array formula GROWTH(R1, R2, R3) for log-log regression is =EXP(TREND(LN(R1), LN(R2), LN(R3))). For example, if we want the y value corresponding to x = 26, using the above model we getĮxcel doesn’t provide functions like TREND/GROWTH (nor LINEST/LOGEST) for power/log-log regression, but we can use the TREND formula as follows: We can also create a chart showing the relationship between ln x and ln y and use Linear Trendline to show the linear regression line (see Figure 3).Īs usual, we can use the formula described above for prediction. We can also see the relationship between x and y by creating a scatter chart for the original data and choosing Layout > Analysis|Trendline in Excel and then selecting the Power Trendline option (after choosing More Trendline Options). ![]() We now use the Regression data analysis tool to model the relationship between ln y and ln x.įigure 2 – Log-log regression model for Example 1įigure 2 shows that the model is a good fit and the relationship between ln x and ln y is given byĪpplying e to both sides of the equation yields The table on the right side of Figure 1 shows y transformed into ln y and x transformed into ln x. It follows that any such model can be expressed as a power regression model of form y = αx βby setting α = e δ.Įxample 1: Determine whether the data on the left side of Figure 1 is a good fit for a power model.įigure 1 – Data for Example 1 and log-log transformation This equation has the form of a linear regression model (where I have added an error term ε):Ī model of the form ln y = β ln x + δ is referred to as a log-log regression model. Taking the natural log (see Exponentials and Logs) of both sides of the equation, we have the following equivalent equation: 2.5%(0.025) in one tail t α/2 = t 24, 0.025 = 2.Another non-linear regression model is the power regression model, which is based on the following equation: Since the population variance is unknown (the variance of monthly returns of Stock A over its entire history, we only have data for the past two years) we will use t statistic.ĭegrees of freedom = 24 – 1 = 23 (two years = 24 months)įor confidence level of 95%, 5% error in both tails, i.e. Compute the 95% confidence interval for the average monthly returns for this stock. The stock has a mean return of 2% and a standard deviation of 8%. You construct a sample of monthly returns of Stock A for the past two years. The confidence interval can be calculated as Since the population variance is known (the standard deviation of all large cap stocks), we will use Z statistic.įor confidence level of 99%, 1% error in both tails i.e. Construct a 99% confidence interval for the average return all large-cap stocks for the past year. Assume that the average returns for all large-cap stocks in the economy follow a normal distribution with a standard deviation of 3%. The average returns of these stocks for the past year is 12%. You take a random sample of 100 large cap stocks. Use the following formulae to calculate the confidence interval: ![]() Refer to the table below and select t statistic or z statistic as per the scenario. ![]() To calculate a confidence interval for a population mean, follow these steps: Concept 12: Calculating Confidence Intervals ![]()
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