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A monthly increase of 176,000 for private nonfarm payrolls in January is expected in Friday's update from the Labor Department, based on The Capital Spectator's average econometric forecast. That's slightly higher than December's gain, but is moderately below a pair of consensus forecasts.

Here's a closer look at the numbers, followed by brief summaries of the methodologies behind The Capital Spectator's projections:

VAR-8: A vector autoregression model that analyzes eight economic time series in context with private payrolls. The eight additional series: ISM Manufacturing Index, industrial production, aggregate weekly hours of production and nonsupervisory employees in the private sector, the stock market (S&P 500), real personal income excluding current transfer receipts, real personal consumption expenditures, spot oil prices, and the Treasury yield spread (10-year less 3-month T-bill). The forecasts are run in R with the "vars" package.

ARIMA: An autoregressive integrated moving average model that analyzes the historical record of private payrolls in R via the "forecast" package.

ES: An exponential smoothing model that analyzes the historical record of private payrolls in R via the "forecast" package.

R-1: A linear regression model that analyzes the historical record of ADP private payrolls in context with the Labor Department's estimate of US private payrolls .The historical relationship between the variables is applied to the more recently updated ADP data to project private payrolls. The computations are run in R.

Source: U.S. Nonfarm Private Payrolls: January 2013 Preview