The think tank said the UK's economic recovery had become "entrenched".
The estimates are broadly in line with those of other forecasters, including the UK's Office for Budget Responsibility.
NIESR also said it expected the unemployment rate to fall below 7% before the end of the year.
Jobless figures released last month showed that the unemployment rate fell to 7.1% in the three months to November.
Last year, the Bank of England said it would consider raising interest rates from their current historic lows if the unemployment rate fell below the 7% threshold, though it has since played down expectations of rate rises in the near future.
NIESR's forecast follows similar raised UK growth forecasts from the International Monetary Fund (IMF) and the Organisation for Economic Co-operation and Development (OECD), which are also increasingly optimistic about the UK's economic prospects.
'Remarkable performance'
Falling unemployment and rising house prices have helped to encourage consumers to spend more, fuelling the recovery.
More sluggish sectors of the economy such as construction are also now showing signs of strengthening.
But concerns remain - particularly levels of business investment, which remain low, and stagnant wage growth which means prices are continuing to rise faster than many people's salaries.
"The UK's economic recovery is entrenched," the NIESR said in a statement. "Above trend growth returned in 2013, while the remarkable performance of the labour market persists."
"We expect consumer spending to remain the key driver of recovery in 2014 and 2015, supported by continued buoyancy in the housing market."
It added that the rapid fall in unemployment seen in recent months had "raised questions over the credibility" of the Bank of England's forward guidance, which saw 7% unemployment as an important threshold.
NIESR said it was forecasting a rise in interest rates as early as the second quarter of 2015, though this is expected to be a year after the 7% threshold is breached.
The Bank of England opted to keep its benchmark interest rate unchanged at 0.5% again on Thursday. The rate has been at the historic low since March 2009.
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