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Basic Statistics and Data Analysis
The Method of Moving Averages
The method of moving averages are of two types: Simple Moving Averages Weighted Moving Averages Simple Moving Averages If the observed values of a variable $Y$ are $y_1, y_2,\cdots, y_n$ corresponding to the time periods $t_1, t_2,\cdots, t_n$, respectively, the $k$-period simple moving averages are defined as \begin{align*}a_1 &= \frac{1}{k} \sum_{i=1}^{k} y_i\\a_2 &= \frac{1}{k} \sum_{i=2}^{k+1} y_i,\\a_3 &= \frac{1}{k} \sum_{i=3}^{k+2} y_i \\\vdots &= \quad \vdots\\a_m &= \frac{1}{k} \sum_{i=m}^{n} y_i\end{align*} where $a_1, a_2, \cdots, a_m$ is the sequence of $k$-period simple moving averages. That is, the $k$-period simple moving averages are calculated by averaging first $k$ observations and then repeating this process of averaging the $k$ observations by dropping each time the first observation and including the next one. This process is continued till the last $k$ observations have been averaged. For example, the 3-period simple moving averages are given as: \begin{align*}a_1 &= \frac{1}{3} (y_1+y_2+y_3) = \frac{1}{3} \sum_{i=1}^{3} y_i\\a_2 &= \frac{1}{3} (y_2+y_3+y_4) Read More …
Method of Semi-Averages
The secular trend can also be measured by the method of semi-averages. The steps are: Divide the time series data into two equal portions. If observations are odd then either omit the middle value or include the middle value in each half. Take the average of each part and place these average values against the midpoints of the two parts. Plot the semi-averages in the graph of the original values. Draw the required trend line through these two potted points and extend it to cover the whole period. It is simple to compute the slope and $y$-intercept of the line drawn from two points. The trend values can be found from the semi-average trend line or by estimated straight line as explained: Let $y’_1$ and $y’_2$ be the semi-averages placed against the times $x_1$ and $x_2$. Let the estimated straight line $y’=a+bx$ is to pass through the points ($x_1$, $y’_1$) Read More …
The Method of Free Hand Curve
The secular trend is measured by the method of the free-hand curve in the following steps: Take the time periods along $x$-axis by taking appropriate scaling Plot the points for observed values of the $Y$ variable as the dependent variable against the given time periods Join these plotted points by line segments to get a historigram Draw a free hand smooth curve (or a straight line) through the histogram In this method we draw the given times series data on graph paper, then we draw a free hand trend line through the plotted graph according to the trend of the graph. Then we read trend values from this free-hand trend line. It is generally preferred to use a curve instead of a straight line to show the secular trend. Merits: The free-hand curve method is a simple, easy, and quick method for measuring secular trends. A well-fitted trend line (or Read More …
The Secular Trend
For the estimation of the secular trend of a time series, the most commonly used method is to fit a straight line $\hat{y} = a+bx$, an exponential curve $\hat{y}=ab^x$, and a second-degree parabola $\hat{y}=a +bx+ cx^2$, etc, where $y$ is the value of a time series variable, $x$ representing the time and all others are constants (the intercept $a$, and the slope $b$). The method of least squares is a widely used method to determine the values of the constants appearing in such an equation. It is used For the purpose of prediction (or projection) into the future The detrending process (removal of trend) in a time series for studying other non-trend fluctuations. It is used for historical description The secular trend can be represented either by a straight line or by some type of smooth curve. It is measured by the following methods: Method of the free-hand curve Method Read More …
Consequences of Heteroscedasticity
The following are consequences of heteroscedasticity when it exists in the data. The OLS estimators and regression predictions based on them remain unbiased and consistent. The OLS estimators are no longer the BLUE (Best Linear Unbiased Estimators) because they are no longer efficient, so the regression predictions will be inefficient too. Because of the inconsistency of the covariance matrix of the estimated regression coefficients, the tests of hypotheses, (t-test, F-test) are no longer valid. Learn about Remedial Measures of Heteroscedasticity
GM Statistics
Percentages, Fractions, and Decimals
Percentages, Fractions and Decimals are connected with each other. We often see the phrases like up to 75% off on all items 90% housing loan with low interest rates 10% to 50% discount advertisments These are some examples of percentages. Suppose, there are 200 students …
Significant Figures: Introduction and Example
Rounding of numbers is done so that one can concentrate on the most important or significant digits. For example, consider a flat priced at $285500. A rich man might think in hundreds of thousands of dollars. To a rich man, it is easier to think …
Use and Application of Ratio
The ratio is used to compare two quantities of the same kind. Consider in a group of 45 people, 15 of them are females. We can compare the number of males and the number of females in the group in two different ways as, There …
Algebra Introduction
We work with numbers in arithmetic, while in algebra we use numbers as well as Alphabets such as A, B, C, a, b, and c for any numerical values we choose. We can say that algebra is an extension of arithmetic. For example, the arithmetic …
Absolute Error of Measurement
Absolute error of a measurement is the difference between the measured value of an object and its true value. When we take the measurement of an object, it is possible that the measured value is either a little more or a little lower than its …
