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█ CONCEPTS
In his article, Ehlers explains that the average price in a time series represents a statistical estimate for a block of price values, where the estimate is positioned at the block's center on the time axis. In the case of a simple moving average (SMA), the calculation moves the analyzed block along the time axis and computes an average after each new sample. Because the average's position is at the center of each block, the SMA inherently lags behind price changes by half the data length.
As a solution to removing moving average lag, Ehlers proposes a new projected moving average (PMA). The PMA smooths price data while maintaining responsiveness by calculating a projection of the average using the data's linear regression slope.
The slope of linear regression on a block of financial time series data can be expressed as the covariance between prices and sample points divided by the variance of the sample points. Ehlers derives the PMA by adding this slope across half the data length to the SMA, creating a first-order prediction that substantially reduces lag:
PMA = SMA + Slope * Length / 2
In addition, the article includes methods for calculating predictions of the PMA and the slope based on second-order and fourth-order differences. The formulas for these predictions are as follows:
PredictPMA = PMA + 0.5 * (Slope - Slope[2]) * Length
PredictSlope = 1.5 * Slope - 0.5 * Slope[4]
Ehlers suggests that crossings between the predictions and the original values can help traders identify timely buy and sell signals.
█ USAGE
This indicator displays the SMA, PMA, and PMA prediction for a specified series in the main chart pane, and it shows the linear regression slope and prediction in a separate pane. Analyzing the difference between the PMA and SMA can help to identify trends. The differences between PMA or slope and its corresponding prediction can indicate turning points and potential trade opportunities.
Users can customize the source series, data length, and plot colors via the inputs in the "Settings/Inputs" tab.
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NinjaTrader: March 2025
In “Removing Moving Average Lag” in this issue, John Ehlers presents the projected moving average (PMA) and some accompanying indicators. Several of the indicators discussed in the article are available for download at the following link for NinjaTrader 8:
Once the file is downloaded, you can import the indicator into NinjaTrader 8 from within the control center by selecting Tools → Import → NinjaScript Add-On and then selecting the downloaded file for NinjaTrader 8.
You can review the indicator source code in NinjaTrader 8 by selecting the menu New → NinjaScript Editor → Indicators folder from within the control center window and selecting the file.
